Confusing Analysis
If you refuse to play Kg2 and instead move the knight to c7, black will be able to launch a fearsome attack against your king with his queen and two knights, with several mating-threats.
The most natural defense against Nxf3 would be kg2, but that is easily countered with Nh4 and thereafter Qe4 and you'll need to divert all you energy towards escaping a mate.
The other possible move, Kh1 is countered with Qd6, threating both mate and direct capture of your knight. A lazy opponent might choose the latter, while a hungry one would keep harassing your king and develop serious pressure against it.
Try playing this sequence as black, acting all your energy towards the king. Don't forget that black has two horses, after Ne4 the other horse will be a hepful tool in the crusade against the king.
I think the best way to understand the position is to play it out and really try to use blacks advantages. But to give a strategic explanation I made this picture:
In this picture, red represents squares that are not defended by any piece (except the king) and green represents squares that are only defended by pawns (that are very vulnerable, f3 as we know won't last long in your scenario). What is really striking in this picture is that the second rank is totally undefended against an attack, while black has a very mobile queen just around the corner with the support of two knights. Kg2 makes it very difficult for black to take advantage of this cause all pawns then are defended, and neither the knights or the queen have any good squares to launch an attack from. Without that move, Nxf3 gives black a chance to get in, remove the pawns and with a supported queen put serious pressure on the king.