It's a harder question than it might seem. For me, obvious reasons leap to mind, mainly the fact that I'd probably go crazy if I didn't. Of course, that's just me, and still but one aspect. Seriously, though, I think I owe my life to meditation and eastern religion, back in my late teens. It was never an immediate fix, hardly a proximate one, I think it was like that one ray of light finding a chink in the box, that reminds you that the darkness has a limit. I can't give meditation full credit, but it was part, it helped me take myself less seriously.
Tangental to the above, there are real, measurable benefits to meditation. A slew of studies have been done proving that meditation lowers stress hormones and blood pressure, improves the immune system, aids digestion, and may even help with weight loss. Other studies with a more psychological bent have shown that it improves one's ability to focus, increase productivity, develop intuition, and so on.
As for that, I have long enjoyed the fact that meditation seems to help me be more creative. Somehow by reaching down into the Void, however briefly (for I'm not, I'll admit, a very 'good' meditator. My monkey mind is still pretty active), I come out with a fistful of inspiration. Not always immediately, but somehow by tuning into... well, I don't know, the underlying thrum of existence, even for a couple seconds, it has a ripple effect, a lasting one, carried away from the session. I don't know much about the act and process of creativity, it is a mysterious thing; but perhaps it lets me get out of my own way, like a crack in the dam where the water leaks out.
Another benefit would be the way it can enhance or build up your compassion. It helps expand your sense of self to include others, and ultimately, it would be hoped, the entirity of existence when you attain Buddhahood. Of course, for most of us, we can simply accept and appreciate even minor growth here, to help improve relations with those we live and work with, which is definitely a good thing on its own, and has its own ripple effect.
So. Besides just giving you a way to ground yourself down from the scary, annoying, anxious, stressful, angry, depressed, selfish, and/or restless mind, allowing you a window of peace in a mental world which is, being human, the major part of our world, besides lowered stress hormones and blood pressure, more creativity and love for others, why do this ridiculous thing, sitting on the floor with your legs twisted up, doing nothing useful or productive?
Well, the simple answer is the basic wonder of just spending 15 or 20 minutes alone with the sheer improbability of your own existence. It is said in Buddhism that to meditate to acheive any of the above benefits, the health, the psychological, the creative, it's all fine but it won't get you Nirvana or release. Meditating with expectation or anticipation is the opposite of meditation. Meditation is simply how a Buddha sits. When he sits, he sits, when he walks, he walks. There is no purpose to meditation.
Easy for a Buddha to say. For the rest of us, even if we are all already Buddhas and just don't know it, the point or purpose is to attain that clarity of understanding that we too, may simply sit when we sit, rather than battle a mind that won't let us just sit, but suddenly we get all itchy, suddenly we get all fidgety, suddenly we know there are a million better things to be doing, and spend 95% of our 15 minutes thinking about other things, and end up not being present at all. Oh well.