This is a pretty big subject with a lot of anecdotal evidence and a growing body of scientific research that both supports and refutes some of what I will tell you below. Like most things, the answer to this question really needs to start with a different question: What do you want to get from a massage? Because the answer to this question depends a lot on whether you see a massage as a once-a-year luxury, sort of a pampering treatment, or as a once-a-week, integral part of your training regime, or somewhere in between.
But, I also know that most people don't want long-winded, intricately detailed answers, in most cases. So, for now, I will answer this question about massage with my personal experience and the experience of my friends, clients and fellow athletes, and then you can look at some of the links at the bottom to learn more - or get in touch, ask me the same question in person, and we can go into as much detail as you like!
Anecdotally, in my own personal experience of receiving massages and giving massages, relaxation seems to be one of the most immediate benefits of a good massage. For most of us, there is just something really nice about putting yourself in the hands of a well-trained masseuse and receiving some expert attention that relaxes you and soothes you. And, there are all kinds of objective physical cues that prove just how relaxed you can feel when receiving a massage: your heart rate lowers and lowers throughout the session, you might even feel sleepy and nod off for part of the session, and your stomach may growl (a sign that your parasympathetic nervous system is taking over from your sympathetic nervous system is that your digestive system gets more active).
If you think about it, a lot of the aches and pains we feel in our bodies that drive us to want to go and get a massage in the first place are either caused by stress or are causing us to feel more stressed. So, going and doing something that relaxes us and reduces the associated anxiety is going to help alleviate some of those muscular problems, even if only for a little while.
This is also why it is important for you, when you go to get a massage, to be quite clear about what you like, what you don't like, what you hope to get out of the session, and to communicate that to the massage therapist. For example, if you don't like incense in your massage room, no matter how many other people like it and say it relaxes them, it is important for you to say that you don't like it, or else you will probably lie there the whole time feeling quietly annoyed that they are burning incense, and not getting the full benefits of relaxation that you came for. Obviously, each masseuse is different, each location is different, and they can't always provide you with everything that you would like, but it is good for you to think about and vocalize what you would like in your session. Be a partner in your own relaxation and help the massage therapist to help you the way that works best for you.
Raised Body Awareness
I have experienced something very interesting when I have been to a really, really good sports massage therapist who really understands the kind of strains I might put my body through: I have discovered hidden connections and related issues in my body that I didn't know were there.
For example: your left knee hurts when you are running and so you tell your masseuse that you want them to look at your left knee. You have a little conversation about when it hurts, when it stops hurting, how often, how intense, what you have tried before, etc. And then, at some point in the massage session, they look at the connected areas - the calf muscles, the thigh muscles, the ligaments and tendons all around the knee, the front and back of the leg, over at the right leg, too, and maybe even up into the hips, glutes, lower back, and beyond. And, with a really good massage therapist, you discover other areas of soreness and tension that are connected in some way, and you come away with a greater understanding of how that tightness in your hips and glutes is probably contributing to that pain in your knee. Even better, you might even come away with a greater understanding of how you can do something about it, a little bit every day, that will help fix those problems and prevent them from recurring.
Just by having a trained and knowledgeable professional put their hands on you and identify these connected and related areas, you can get a really lovely, deeper understanding of how your body works. I love it when that happens to me in a massage session, and I love helping my clients have that same experience when I give massages, too. I would argue that, long-term, this is the real #1 benefit of a good massage - a better understanding of your body and how it works.
Injury recovery and injury prevention
This is sort of a corollary to the point above, the idea that massage helps raise awareness of your body. The fundamental principle underlying massage is that a well-trained masseuse who understands human anatomy very well can identify muscles and muscle fibres under your skin, identify areas of tension and bring them to your attention. At the very least, if you can identify these areas of tension, and figure out the root causes, you can better support the recovery process. And, by working with a sports massage therapist regularly, you can identify minor aches before they becomes a serious injuries.
As part of my training over the years, we have always been told that massage will also increase circulation to the area being treated, therefore delivering more oxygen and nutrients to those muscles, and facilitating the removal of lactic acid and other waste products from them. We have also been told that we can manually facilitate the removal of adhesions in the damaged or overused muscle tissues and help realign the muscle fibres to make them stronger and restore optimal range of movement. Some research has come along to question these assumptions about massage, and still other research backs this up. As you might imagine, scientific sampling (taking muscle biopsies, blood samples, other tissue samples) is much more arduous and painful than asking an athlete to "rate their feelings" or "describe the aches and pains", and so gathering scientific data to support these claims is still sometimes hit and miss. So, I am still interested in reading more about this and better understanding the science of how massage can help people recover from injuries and prevent them from recurring.
However, we can all agree that, if a good massage can satisfy the first two points above (Relaxation & Raised Body Awareness), then it will certainly help athletes to recover more comfortably and understand their bodies better to avoid recurrence of injuries in the future.
Here are some other sources of information I have seen on the subject, some of which will back up what I am saying, some of which will call it into question, and some of which may encourage a whole different range of thinking on the subject, too: