Starting with some really sound, useful tips from another source:
Now, some full disclosure from me: from about the age of 10, I have been a keen martial artist, starting with Judo, then Karate, then Tae Kwon Do, then Wing Chun, then T'ai Chi, and several other arts and styles along the way. In almost every class I ever took, there was a strong emphasis on stretching. And, from the start, I enjoyed being very flexible, as a shorter-than-average kid, being able to get my leg straight up in the air, being able to kick as high as a taller opponent's head, and being able to execute complex, flexible, spinning and jumping maneuvers. Especially when I was studying Tae Kwon Do during my college days in the 1980s, we stretched A LOT- we stretched really, really thoroughly at the beginning of every class, we stretched at the end of every class, and we stretched quite a bit in between. The idea (which I will go into further below) was that "loose is fast" and we all wanted to be as fast as we could be. And, I have to say, we learned to be pretty damn fast through the guidance of a guy who was regional and world champion in his weight division.
Then, after getting full-time jobs in offices and having children, my martial arts career slowed down and I eventually found my way into the world of triathlon. I encountered runners and cyclists and swimmers who all had very different relationships with stretching that sometimes baffled me. I really was mystified at how anyone (especially my hard-core runner friends!) wouldn't agree that being loose and flexible (like being able to touch your toes without bending your knees) is a de facto good property for your body to have. Of course, they laughed at me and pointed out study after study after study that showed runners and cyclists enjoyed little or no benefit from stretching, and they smugly went about NOT stretching to their heart's content. That didn't deter me and, to this day, I remain a big fan of stretching.
I also happen to love yoga. A good yoga class led by a good yoga instructor (for example, friend and fellow triathlete, Gail Wilkinson, https://gailwilkinsonblog.wordpress.com/yoga-for-athletes/), can also show you how flexibility and strength go hand-in-hand. You can be supple AND strong, relaxed AND powerful, at the same time. These are sensations I have had in good martial arts sessions, and they are feelings I also try to bring into my triathlon training and coaching. They inform a lot of what I teach and practice in swimming sessions, and these ideas are steadily making their way into running and cycling circles, too.
So, I have a relationship with stretching that will certainly bias me towards how I answer this question, and I thought you should know that from the start. Having said that, there are also some pretty simple, widely accepted answers to the question about the benefits of stretching.
Static and Dynamic Stretching
There are differing opinions out there, and I personally have had a lot of varied experience with stretching that flies in the face of current received wisdom in sports science and coaching circles (see my comments about stretching in martial arts classes, and how that differs from the recommendations below). But, the main recommendations that most people can agree upon are these:
Before activity, it is best to warm up with some dynamic stretching. This is usually a more rapid stretching movement that you don't hold for very long and takes you through more than a full range of motion for the activities you are about to undertake (e.g. swinging your legs back and forth from the hips and knees before you go for a run, swinging your arms around in circles before going for a swim, etc), effectively warming up your body before the planned activities get started in earnest.
After activity, it is best to cool down with some static stretching. These are stretching postures that aim to stretch out the muscles you have just been using (e.g. calf stretches after running, shoulder stretches after swimming, hip stretches after cycling, etc), and they are held for longer periods of 10 seconds or even up to 1 minute, effectively cooling your body down after the activities you have just done.
If you find at any point in your activity that you have a knot or a cramp in a muscle, then it is usually recommended that static stretching can help alleviate these knots or cramps, even if it is not taking place at the time usually recommended for static stretching (i.e. normally only recommend static stretching at the end of the session).
If you apply some common sense, then this makes a lot of sense. If you are about to do something high-impact, like running, then you don't usually want to do lots of static stretching which will lengthen and soften the muscle tissue at the beginning of the session; instead, you want to warm the muscles up, get the body moving with shorter, more active, dynamic movements, get the heart rate up a little bit, and get the body ready for the exercise you are about to do. Then, at the end of the session, it makes sense to slow down, hold the stretches for longer, cool the body down, bring the heart rate down, and ease out of the session when you are done with the main activity.
Range of Motion
You may not think that there is much range of motion involved in the triathlon-related sports of running and cycling. I mean, how much flexibility do you need to be able to run middle- to long-distance on a road or a path, or move the pedals around on a bike? Well...actually, think a little bit wider for a minute...and imagine the flexibility needed to swing your leg over the saddle of your bike quickly and easily, or jump over a log on a trail, or suddenly change direction to avoid an obstacle on the street. There is a range of motion that goes beyond the narrowest limits of a sporting activity, and there are definitely times when I appreciate having more flexibility than your typical runner or cyclist seems to have. Certainly, when it comes to swimming, it is clearly necessary to have a good range of motion through the shoulders and upper torso.
In these cases, it is very simple: stretching your limbs and muscles beyond the anticipated range of motion for the sport you are about to do can't hurt, and indeed it might help. I think back to my martial arts training, trying to stretch my legs up as high as possible, knowing that I was probably not going to do a lot of kicking to the very top of my range of motion, but also feeling a sense of ease and confidence knowing that I could kick that high if I wanted to, arguably made my kicking movements more fluid, speedy, and precise. When going for a run, you may not need to lift your legs up above your waist, but it can be really good to know that you can if you need to (e.g. jumping over or avoiding obstacles in your path).
Warming Up and Cooling Down
Most people don't do enough of this, plain and simple. I will confess that, even with all my love of stretching, all the advice I give to other people, there are plenty of times when I don't do the stretches, dynamic or static, I know I should be doing when I train. I know, I know, I know. It is raining and cold, I want to go for a run, and I have just enough time to do the run I want to do before I have to be back home and get dinner ready. I don't want to take 5 or 10 minutes to warm up, I just want to run. And, I will probably not stop and stretch for 10 or 15 minutes when I get back, cold and wet and wondering what I am going to cook for dinner tonight. I know, I know, I know. I am sometimes a hypocrite in that regard. But, I also know that my training session is much stiffer and slower when I don't warm up before the session, and my body aches more noticeably for a day or so when I don't stretch out afterwards. I feel the differences, right then and there, and for hours and hours afterwards, too. A good stretching routine helps with that, a lot, and makes it more possible for me to train multiple times in one day without compounding aches and strains, as most triathletes will find themselves needing to do, at some point.
Recovering and Rebounding
Following on from the point made above, when you do take the extra time to do the dynamic and static stretches recommended, you will probably also find that your body bounces back more quickly after each work-out. You are less likely to be carrying tension or aches and pains from one work-out to the next, whether the next work-out is the same day or the next day.
It is also a way of being in better touch with your body and its on-going condition. When you take that extra 10 minutes or so at the beginning and end of every session, you are also taking time to check in with yourself, take a little inventory and, over time, notice the little and big changes in your body. Rather than just racking up the miles and hours of training, you are also more likely to pay attention to some of the qualitative dimensions of your training and ask interesting questions of yourself, such as, "Why is my left leg more sore than my right leg at the end of that long bike ride? Is there something imbalanced in my pedaling? Is that something I can work on, and perhaps improve my power on the bike?"
Loose is Fast
These are just my ideas, based on my experiences, my thoughts and feelings - totally anecdotal, not at all scientific:
If you are all tightened up when you do something, anything really, you are going to hold a lot of tension in your body when you do that thing.
If you hold a lot of tension when doing something, you are probably wasting a lot of energy with that tension that could otherwise be used towards accomplishing that activity.
So, in order to be as fast as you can be, you have to try and be as relaxed as you can be, as loose as you can be, saving as much of your energy as you can for the actual activity you are participating in.
So, loose is fast. Tense is slow.
There is a time to tense up, usually relatively briefly, for whatever point of contact or explosive effort is required. For example, in swimming, your arms remain relaxed throughout the recovery phase of the stroke, into the hand entry phase at the front of the stroke, and even through the catch phase where you set the forearm and hand in place. Then, and only then, do you start to tense up and exert more power, once the hand and forearm are in place and the thrusting phase of the stroke begins, then continue accelerating through the stroke to a powerful finish, down as close to the knee as possible.
Similarly, when running, the legs and feet should be relatively relaxed when they recover back to the front of the stride, and even stay pretty relaxed as you bring the foot down, choosing the surface, distance and angle for the foot strike. Then, when the foot makes contact with the ground, it will need to tense up somewhat, absorb the impact, and then get more powerful as your body passes over the foot, then the foot and leg transfer the power to the back of the stride for one last explosive thrust that propels your body forward into the air, and then repeats the cycle. Additionally, in running, we want to stay pretty loose and relaxed in the upper body. You certainly don't want to be carrying much tension in the arms, shoulders, chest or back when running. Let the legs do the vast majority of the work.
Cycling is interesting, and different in this regard from the other sports, because the nature of the pedal revolution is that you want to maintain as smooth, consistent and circular a flow of power as possible throughout the entire pedal cycle. We actually can do a lot of drills and analysis to work closer and closer to a truly circular, even delivery of power throughout each pedal stroke. And, like in running, you want to stay pretty relaxed in the upper body when cycling. Indeed, carrying a lot of tension in the shoulders, arms or hands while riding your bike can be quite dangerous, as that tension can lead to problems steering and braking on the bike.
So, there is a time and a place for tension, a time and a place for relaxation. Other people have said it better than me:
"Float like a butterfly, sting like a bee." -- Muhammad Ali
"Tensing up robs you of speed; relax and let go, then you will be ten times faster." -- Bruce Lee
"Swimming requires relaxation, not rigidity. Take time to play around: splash, breathe, close your eyes, lie on the bottom, float, blow bubbles, go limp, do roly-polys! Learn to feel – not fight – the water." -- Chrissie Wellington
"You need to make it a regular part of your training - stretching & cooling down. Simple as." -- Bradley Wiggins
"Easy is light, and light is fast. Run easy and light to run faster." -- Micah True aka "Caballo Blanco", ultra-runner