Physicist: Isn’t that weird?
The name “quantum mechanics” comes from the fact that, at its most base, quantum mechanics requires all particles and energies to come in discrete (one might say “quantized”) packets. At some point a bunch of physicists starting asking awkward questions like; the matter is quantized, the energy is exchanged in quantized packets, so why do we assume the force is smooth and continuous?
Compounding this awkward line of questioning was the fact that photons were already known to carry electromagnetic force. Literally, photons are little oscillating bits of electric and magnetic fields, which is exactly what the electric and magnetic forces are. So the next obvious question was “do the other forces have ‘force carriers‘?”
You’re damn right they do. Photons for electromagnetism, W and Z bosons for the nuclear weak force, and gluons for the nuclear strong force. There’s every force but gravity! Each of the carriers were predicted by the (then new) study of “quantum field theory”, and have since been observed. The theory itself is gorgeous and works beautifully. In fact, it barely makes sense to think of anything in the universe (including space) as not being quantized.
So, some physicists are looking for evidence of the existence of gravitons (the gravity particle), because it would really tie things together nicely. There are a couple drawbacks however… In order for something to be detected it has to do something. Gravity is a really, really weak force, and a graviton is the smallest amount of that force that can exist. Most physicists have already given up any hope of detecting the graviton directly, and instead are looking at extremely indirect methods. The drawback there is that the graviton (if it exists, and if our theories hold up) is a very strange particle, and is described using amazingly nasty math (even more nasty than the normally nasty math of quantum field theory). So it’s difficult to even figure out what those indirect methods should be.
To actually answer the question: some physicists are looking for the graviton because it “fits”.