h7 is a less active square -- no reason to bury the bishop.
Alternatives to moving the bishop would be queenside play (black will be going for c5 and d4 pressure sooner or later I'm guessing). But black is not ready for the position to open (development and needs castle). Therefore to open up squares for the bishop and continue development black clears the f5 square. Because again going for his natural play is too soon as it opens lines before development is complete.
Well j-r-p, I too wonder why, if black decides to retreat his bishop, why he doesn't just put it on h7. It's safer there; not exposed to any Nh4 or h4-h5. I don't know if those things are realistic, but on h7 you can forget about them entirely. It seems to control the same main diagonal as the bishop on g6. There must be some reason the grandmasters pick ...Bg6 (or maybe they pick both), but I wouldn't be able to tell you what that reason is.
Anyway, the general idea is as wafflemaster described. Black improves his pieces before opening the position up, in particular by clearing f5 for a knight.
The difference between g6 and h7, I would guess, is only tangible at a very high level. I just happen to prefer keeping more diagonal options open i.e. from g6 it has the option of shifting to h5.
Is that diagonal ever going to be useful in the game? No idea. Does keeping it on g6 accelerate any ideas of kingside attack by white? No idea. But if you told me GMs prefer g6 to h7 that's what I'd guess anyway 
I don't think grandmasters know the answer themselves :)
I checked megabase and found that in one game black used the bishop on g6 to support the advance h6-h5, probably intended to obstruct g2-g4.
Thanks all. In summary, Bg4 instead of Bh3 mainly because:
- it gives you flexibility to later play Bh5 if you want
- White's Nh4 wastes time because you simply respond Bh3 anyway, and now he has a knight on the edge of the board
There's other minor advantages like later supporting h4, but those two seem most important.
Estragon, I have to say, I'm a bit surprised that ...Bg6 is better partly because white has options of Nh4 and h4-h5, not despite it 
But isn't it good to have those extra options? Unless of course you're playing a psychological trick. But on objective grounds, that's my conception of safer -- the bishop on h7 is, of course, actually defended less times than one on g6, but it's simply harder to get to -- we wouldn't consider, for example, the h8 rook to be the least safe piece on the board just because it isn't protected -- there is no way to get to him 
So I agree with your idea that we want to be elastic, but that's part of why I liked ...Bh7 -- it sort of makes white's position have less flexibility by taking out those two options. Just as ...Bh5 may not always be good, Nh4 need not always be bad.
But perhaps, Estragon, from your experience, you find that it's more likely that black will want to play ...Bh5 or ...h5 than it is for white to want to play Nh4 or h4-h5 in the case of ...Bg6? That may be the case, but on more general grounds I disagree with you about what is flexible here -- ...Bh7 doesn't increase black's flexibility, but it does decrease white's flexibility.
Another reason for Bg6 is Nh4 becomes wasted when black responds with Bh7. Now at some point, white has to return the knight from the rim.
But if you play ...Bh7, Nh4 wouldn't do anything at all
So I think the answer more has to do with how powerful black's extra possibilities of ...h5 and ...Bh5 are, and if they outweigh white's possibilities of Nh4 and h4-h5 in the case of ...Bg6. Perhaps black's options of ...h5 and ...Bh5 are more threatening than white's h4-h5 and Nh4 ideas.
And if that's true, why not bring the bishop all the way to Bh7 assuming there's a hole at h7?