This game puzzled me, and I figured it'd be worth analyzing for my benefit.  How could I have won this?   Where were my and my opponents' major blunders?

Also, If anyone could explain what the heck 1.h3 is good for, then I'm all ears!

1. h3 is utter crap. There is nothing to explain.

2. You need to protect your pieces better. In the middle game you ended up with one pawn down wich creates problems later in the endgame

3. You should not trade if you are a pawn down. It turned out ok because your opponent messed up and gave away his lead to a drawish rook pawn ending.

4. There is nothing you could have done in the end. Rook pawn like that draws if the enemy king gets near the corner. Still, you should maybe look up what opposition is. It helps in king and pawn endgames.

1.h3 ,I think, is just a wasted move.

One thing I noticed about the character of this game.  The both of you were content to trade pieces.  That in itself will point a game toward a draw.

h3 couldve been a 'safe move' that they used so they could see what type of open youre going for. itsd not really strong in any regard, but for a defensive player (which really, this guy didnt seem to be) it could make sense.

at move 20 you should have played r-d3.  you "trade off pawns and it leaves his pawns fractured.  You actually end up winning the "new" e pawn after fxe and rxe.  then you can even put pressure on a new point with n-e5.  That seemed to be the  point that jumped out at me.    

Baffled? You played the opening beautifully up to move 6. His 1.h3, as everyone has mentioned, is pretty much the same as not moving at all, it throws away his first move -- in effect you are the one playing White... so you played in effect a Queen's Gambit... so we have a Queen's gambit with colors reversed. When he played 6.Nc3 you needed to see (calculate) that 6...Nf6 was going to lose a pawn (which is what happened). You had better: 6...d5 looks promising IMO.

Look at the position after 12.Qc2 ... what do you see? You have castled and developed your minor pieces; his Kingside is completely undeveloped and his King is in the middle. You needed to play aggressively and actively, mobilize your rooks and get some lasting advantage from this temporary one. If you play slack, he gets his pieces out and it's an even game.

12...Qf5?  horrible... all this does is give him a free tempo. He trades, you recapture, and now it's _his_ move. What have you gained? Nothing, you've just handed him a free move, and exchanged your actively posted Queen for his relatively passively posted one. 12. Rad8 fighting for control of the open file and keeping his king in the center is much better, I think. 

15.Rxd5 ... this is nonsense on his part, and his 16.f3? is bad althouh at least he's playing with an idea (shutting down your bishop's diagonal)  But you make nothing of it. You own the e-file. You have much more active pieces. You must analyze his position, find a weakiness, and attack it! Otherwise your advantage evaporates.

here's an idea... 16...Bb1!? Now if 17.Kxb2? Rd1+... winning back your piece and pinning his miserable N and R. Your one rook immobilizes both of his remaining pieces... neither can move without losing the other! Now you can manuever your N freely to continue attacking. 

Instead you play e4? You are helping to free him from his self made prison. 

For a player with a rating under 1500 you are doing a GREAT job of not hanging pieces, not making big material losing blunders. That's a first important step! But the next step is not making _simple_strategic_ blunders. I'm not talking deep strategy here, I'm talking abc's of strategy. Trading pieces only when it's to your advantage. Not helping him out of a cramped position with pointless 'attacks' and pointless pawn moves.... Your play on his kingside from move 15 on is a study in what not to do. I'm not trying to be mean, just help you improve. You trade and push pawns and just generally free up his position. Honestly he plays even worse than you. It's like he doesn't want to use his pieces, and you have to convince him to.  And you do :(

I strongly recommend you start studying Jeremy Silman's Reassess Your Chess, there's a lot there, you don't have to master it all in a day or a year, but dip in, study the stuff that interests you... I think it will help you play better chess.

*edit* An important note:  -- my idea: 16...Bb1!? Now if 17.Kxb2? Rd1+ ... that is great, but of course it's predicated on White playing a blunder. One must never make a move that depends on the opponent making a bad move. The real point of 16....Bb1!? is 17.a3 Ba2 with the idea of 18...Bb3 which gets control of the d1 square for your rook to penetrate his position.

It's nice to play a move which sets a trap for the obvious reply, but it must be sound against the best reply... I believe 16...Bb1 meets that criterion.

The first thing that jumped out at me was at move 25. See the diagram below and be sure to look at the Move List.

Yes. Ryannolt's suggestion works with the same weak e1 square and pin on the N and R that I suggested targetting a few moves earlier. It's like White just doesn't understand he needs to use his pieces! Ya gotta punish him for his sins! After 26.b4? you still had Rf1.

Remember this old saying: The pin, she win!"

Thanks so much for all your help!  I appreciate the ideas, suggestions to help further develop my skills, (especially JG27Pyth) and general demonstrations of "how to think".

Chess.com and many people on it are awesome!

Matalino, yes, that was exactly my point. Just click the Move List and you'll see a variation in blue. Those are the moves I suggested.