I can think of no explanation for Black's 16th move pawn-grab other than the one that I provided for another opponent's 11th move swipe, about a week ago: he simply did not see that the pawn was protected.
perrypawnpusher - eduborio
blitz, FICS, 2011
1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Nc3 Bc5 5.Bxf7+
The Italian Four Knights Jerome Gambit.
5...Kxf7 6.Nxe5+ Nxe5 7.d4 Bxd4 8.Qxd4 d6
9.0-0 Be6 10.f4 Nc6 11.Qd3 Rf8
Preparing to castle by hand.
12.f5 Bd7 13.Bg5
Instead, 13.Qc4+ first, as in perrypawnpusher - hklett, blitz, FICS, 2010 (1-0, 18), was more accurate.
13...Ne5 14.Qg3 c6 15.Rad1 Qb6+ 16.Kh1
16...Nxe4
Inexplicable, as are Black's next two moves as well. Perhaps he lost interest in the game.
17.Nxe4 Nc4 18.Nxd6+ Kg8 19.Nxc4 Black resigned
Black is not only down a piece and a pawn, his Bishop is difficult to defend – and he faces a mating attack on his King with Bh6 (a move that White could have played instead of capturing the Knight).