Well, in fact white doesn't quite have a "clear advantage" in this position. Black should continue with Ke7 here, after which white must retreat his bishop (usually to d5) otherwise he loses a piece after ...h6. Now black normally follows up with either 6...d6 or 6...Qe8, intending ...Qg6 (it doesn't really matter which one goes first since white can't win the rook after 6.Bd5 d6 7.Nf7? Qf8! 8.Nxh8? Bxf2+! [9.Kxf2?? Nxe4++!] 9.Kf1 Bg4 bags the queen). Then black can develop his light-square bishop (usually to g4) and put his rook on the semi-open f-file, which gives him good play on the kingside.
As you see, all black's developments are very natural and very rapid. White, on the other hand, has to move his bishop and knight several times to keep the extra pawn. Black's king would actually be fairly safe in the center (unless you play inapprioately). Overall, white should be a little bit better in this position, but black's lead in development, control of center, and activity on the f-file are good compensation for the pawn. Just for your information, Beliavsky tried this opening against Karpov in 1983 when Karpov was at his peak, and drew the game. In 1991 he tried it again against Anand and managed to win after Anand's blunder. So, this line is quite playable at amateur level, and maybe at grandmaster level, too.
These are some general guidelines. If you want in-depth analysis, you can get the CD lecture, "The Traxler Counterattack", by Dan Heisman. However, I strongly recommend that you not worry too much about the specific variations, since the middlegame and endgame are what amateurs should focus on, not openings.
C57: Italian Game: Two Knights Defense, Traxler Counterattack
How does it continue... I'm alright if white play Nxf7, but this one, I have trouble with.
The blacks king moves aside and white castles to safety. Does white not get a clear advantage hear. I can't go very deep in the game explorer, being a free-member. Can someone explain the most logical continuations.
TY