There's plenty of links that will tell you yoga is good for flexibility, that it can be a deep spiritual practice, and it has healing effects. All of this is true, but none of it explains why or how. Yoga doesn't work just by making you more stretchy. And you don't have to be flexible to gain benefits from yoga.
I know a lot of my friends don't know what yoga's really about, I'd wager even a lot of people who do yoga don't know (I know I didn't really know until I'd been taking class for a few years).
My ex (who is still a good friend of mine) and I were talking about yoga once, because I'd constantly tried to get him to come with me through the entirety of our relationship*. I thought it would help him SO much and I didn't understand why he was unwilling to try something beneficial. Finally, one day, he said to me, "Okay. Tell me how yoga works. And do it without using words like 'spirit' and 'flow' and 'energy' or any weird hippy-dippy jargon."
And I said, sure thing.
How Yoga Works (Without the Hippy-Dippy Jargon):
Yoga works by cutting off the blood flow to a given part of your body during the posture. When you are in the posture, a tourniquet effect is created, damming up the blood in the area you are compressing, while simultaneously stretching the muscles in another part of your body.
When you come out of the posture, a reserve of fresh, oxygenated blood flows into the area that was compressed.
In our day-to-day lives, most of us spend all of our time sitting, standing, or lying down. Our blood circulates evenly (hopefully) throughout our system. When you do yoga, however, you are sending fresh blood to every part of your body through a series of postures. This "wakes up" your body. Getting a rush of fresh blood provides that part of your body with more oxygen and nutrients in twenty seconds than it would otherwise get during a normal day. This is why you have more energy after class than you do going in.
It's also why yoga is so good for injuries, because you are safely sending healthy blood to a damaged area, rather than letting it atrophy from non-use.
While this compression is happening, there is another part of your body that is getting stretched out, and this is where you gain your flexibility. Yoga teachers stress form over depth. The reason for this is that it's easy to "cheat" a posture, to make it look good by lifting your hips, leaning one way or another, etc. But by forcing yourself to go only as far as you can with proper form, you are stretching specific muscles that may not get used very often, as well as putting your body in proper alignment for the duration of the posture.
Yoga improves your strength, balance and concentration because in order to hold the postures for very long, you have to focus completely on what you are doing. It also challenges you to learn where to distribute your weight so that you don't fall on your ass. And since yoga doesn't use any special equipment**, you are using your own body strength to hold yourself up, twist yourself around, pull yourself down, and so forth.
Those are the physical reasons behind why yoga works and why it heals. Of course, there is more to the practice as a whole than this, such as the importance of breath, but for your basic summary of how yoga works, I feel the above is sufficient.
One more thing I would like to mention is that there is a commonly held misconception that "Hatha" yoga is different from other "types" of yoga (Ashtanga, Iyengar, Bikram, Vinyasa, etc.). The truth is that ALL of these types of yoga are Hatha yoga. Any yoga where you practice a physical series of postures is Hatha yoga. Unless they study yoga and are aware of the seven other branches of the practice (none of which involve physical postures), what most people in America think of when they hear the word "yoga" is Hatha yoga.
So there you go. Yoga is not just about twisting yourself into funny shapes to impress your friends, or sitting in a quiet room and smiling graciously at your peers while whispering "Namaste". There are real, legitimate reasons why it is a physically beneficial and healing practice. Reasons that can be explained to your fellow humans without sounding like a granola munching hippie. Even if that's what you are. ;)
*This is something I don't do anymore. Trying to get someone to go to yoga when they are resisting is like trying to shove a cat into a toilet paper tube.
**For the most part. There are some schools of yoga that let you use props, like a block or a tension band.