Good question. The oscillation of the tuning fork depends on the springiness of the metal prongs. One prong would be springy, so why not use just one?
The reason is that to work properly the tuning fork has to have a balanced motion. It is normally used held in the hand. If you just had one prong, the energy of the oscillation would very quickly be transferred from the handle to the skin of the hand, and would be lost. The result would be that the oscillation would die away very quickly. In technical terms the hand damps the oscillation.
Actually if you were to clamp one prong of a tuning fork in a vice, it would work just fine. The vice is massive and rigid so not much energy is lost into it.
For the same reason, you can twang a ruler on the edge of a desk but you cannot twang it if you are holding it in your hand!
If you have a tuning fork with two prongs of equal size, they can oscillate with motion equal and opposite to each other - balanced in other words. Because the motion of one prong balances out the motion of the other, there is no motion of the handle. Because there is no mechanical energy going into the handle, no energy can be lost into the hand, so the oscillation lasts a long time.