Two Black pieces are loose: the rook at a8 and knight at h5. Look for a square from which the queen might attack both. The lines to them intersect at e8 (way too hard to reach) and d5 (more feasible—White’s queen can get there in one move). The only problem is that the pawn on c6 both blocks the path to the loose rook and protects d5. We handle this with an exchange, of course, drawing the c6 pawn forward by capturing its protectorate. 1. Nxd5, c6xN and now the problem is in simpler form; you see that d5 is occupied but unguarded and the paths to both rook and knight are clear. Qxd5 suggests itself as a working fork.
But now pause and reflect on Black's reply to Qxd5. One of his forked pieces, the knight on h5, is positioned aggressively. Black might be able to break out of the fork by using that knight to do some damage. Indeed, he might play NxBf4; then if White carries out the fork with QxRa8, Black has Nxe2+, a fork of his own. So to prevent all this White doesn't follow 1. Nxd5, c6xd5 with Qxd5 right away. He first moves that vulnerable bishop off of f4, and he does it with a threat that holds the initiative: Bc7, attacking Black's queen. Notice that the bishop has protection on c7 from the rook back on c1, because Black's c-pawn is gone by that time. Black thus moves his queen—and now White can go ahead with the queen fork on d5, because Black can't use his knight to make a capture in reply; the piece he might have wanted to capture—White's bishop—is no longer there.
If Black is alert, of course, he will not recapture on d5 in the first place; he will let the pawn go rather than subject himself to all that has just been described. As often is the case, the value of the threatened fork is that it permits other, lesser gains by making it too costly for your opponent to recoup his losses. (Here Black can reply to Nxd5 with NxB, removing his loose knight from the coming fork; then White has Nd5xN, keeping the pawn.)
It all starts simply: by noticing loose pieces and asking whether you can make a double threat that includes at least one of them. If you do that, seeing the fork is reasonably easy. If you don't, it's hard.