What are the strategic benefits to compensate for the fact that White's giving up the bishop voluntarily and so early in the game in the below variation? What does White hope to get back out of the opening?
[fen ""]
1. e4 e5 2. f4 exf4 3. Nf3 d5 4. exd5 Nf6 5. Bc4 Nxd5 6. Bxd5
I think Jonathan Garber's comment about trying to gain time against the Black queen after 6...Qxd5 7.Nc3 accurately describes what one motivation for 6.Bxd5 can be, though I also think that Black happens to have nothing to worry about in that regard. While she is temporarily behind in development, she shouldn't have any troubles completing her development, while letting the extra pawn on f4 go and simply enjoying her bishop pair on an open board. Being fuzzy about it, I'd say that all in all, everything evens out here, and we have a quite level game on our hands after 6.Bxd5.
I think another way to look at the reasons for 6.Bxd5 is in "what else?" terms. According to my database, the only more popular move in this position (though it is about 3 times more popular) is the sensible 6.O-O, but after the most popular response 6...c6, Black will be happy to camp out with a nice healthy piece on d5, until White sees fit to exchange it somehow. (This could be the same response to other moves as well, e.g. 6.Nc3 c6 and sit tight.) Perusing through some high-level games, it seems that White not infrequently ends up playing a later Bxd5 anyway (though certainly not all the time).
So, one possible takeaway: the position is already quite level after 5...Nxd5, and the immediate 6.Bxd5 is as good an option as White really has to try and create a favorable dynamic imbalance (of the sort Jonathan Garber mentions), even though IMHO Black has nothing to fear in this instance. One reason the King's Gambit isn't terribly popular at the highest levels - the occasional Nakamura game notwithstanding - is that Black does have some fairly reliable methods of defusing White's more ambitious prospects if she knows what she's doing, and this line for Black seems like an instance of that.
As an example of how things can go well for White, here's Magnus Carlsen using this line (by transposition) against Wang Yue. Around the time of 11.Qe2, we have the sort of balance described above, where White has a lead in development and more central space and control, but Black has safely castled and has nice bishops in an open setting. Then Wang Yue somewhat perplexingly engineers a bishop trade himself, and plays rather passively the rest of the way, letting Carlsen do what he does and make maximum use of the remaining White pluses:
[FEN ""]
[Event "Bazna Kings 4th"]
[Site "Medias"]
[Date "2010.06.17"]
[Round "4"]
[White "Carlsen, Magnus"]
[Black "Wang Yue"]
[Result "1-0"]
[WhiteElo "2813"]
[BlackElo "2752"]
[ECO "C36"]
1.e4 e5 2.f4 d5 3.exd5 exf4 4.Nf3 Nf6 5.Bc4 Nxd5 6.O-O Be7 7.Bxd5 Qxd5 8.Nc3 Qd8
9.d4 O-O
10.Bxf4 Bf5 11.Qe2 $10 Bd6 $6
12.Bxd6 Qxd6 13.Nb5 Qd8 14.c4 a6 15.Nc3 Nd7 16.Rad1 Bg6 17.Qf2 Re8 18.h3 Rc8
19.Rfe1 Rxe1+ 20.Rxe1 c6 21.d5 Nf6 22.Qd4 cxd5 23.Nxd5 Nxd5 24.cxd5 Qd6 25.Ne5 Re8
26.Re3 Rd8 27.Nc4 Qf6 28.Re5 h6 29.d6 Bf5 30.Nb6 Be6
31.d7 Kh8 32.a4 g6 33.Qc3 Kg7 34.a5 h5 35.h4 Rxd7
36.Nxd7 Bxd7 37.Qd4 Bc6 38.b4 Bb5 39.Kh2 Ba4 40.Rd5 Bc6 41.Qxf6+ Kxf6
42. Rc5 Ke6 43.Kg3 f6 44.Kf2 Bd5 45.g3 g5 46.g4 hxg4 47.h5 Be4
48.Rc7 f5 49.h6 f4 50.h7 g3+ 51.Ke1 f3 52.h8=Q f2+ 53.Ke2 Bd3+ 54.Ke3 1-0
Nd5 has the appearance of being a nuisance.
Commented
Nov 2, 2012 at 13:38
6. . . Qxd5 7. Nc3. If Black instead plays6. . . c6White returns the Bishop to b4 with a piece gained and little loss of time. Black will have options on the kingside, but he will be significantly behind in development, making it extremely difficult to capitalize on the (relative) weakness of White's exposed kingside.