The reason for having two prongs is that they oscillate in antiphase. That is, instead of both moving to the left, then both moving to the right, and so on, they oscillate "in and out" - they move towards each other then move away from each other, then towards, etc. That means that the bit you hold doesn't vibrate at all, even though the prongs do.
You might ask why it is that they do that, instead of oscillating in the same direction as one another. The answer is that at first they oscillate in both ways at the same time, but the side-to-side oscillations are rapidly damped by your hand, so they die out quickly, whereas the in-and-out ones are not damped this way, so they ring on long enough to hear them. An excellent illustration of this can be seen in this video of a FEM model of a two-pronged fork, which shows you all the vibrational modes separately. (Hat tip to ghoppe, who posted this video in a comment.)
Having a third prong wouldn't help very much with reducing damping. There are (at least) three different ways a three-pronged fork could vibrate: one with all three vibrating side-to-side in phase with one another, and two where one of the prongs stays still and the other two vibrate in and out. (At first I thought there would be three of this latter type of mode, but the third can be formed from a linear combination of the other two: $(1,0,-1) - (1,-1,0) = (0,1,-1)$.) The vibrational mode in which everything moves in the same direction would be damped by your hand, and some combination of the other two would continue to sound for a while. As Ilmari Karonen pointed out in a comment, there would also be a "transverse" $(1,-2,1)$ mode, where prongs vibrate out of the plane of the fork. This mode wouldn't necessarily have the same frequency as the primary in-and-out mode, so it's probably something we'd want to avoid.
But ultimately there's little reason why, in a three-pronged fork, the vibrations would ring on longer than just two prongs - there would be the same amount of vibrational energy as in a two-pronged fork, but just shared between two or three modes of vibration instead of one.