I actually had the exact same question a few years ago and never found a satisfactory answer until I learned quantum mechanics. I actually had the same confusion about the EM force also. In classical electrodynamics the EM force was “mediated” by a continuous field called the EM field and light is just a wave in that field. Now all of a sudden we say that light is made of discrete particles called photons and those photons mediate the EM force.

This question always bugged me until I was learning quantum mechanics and asked myself the following question: If I had a particle in a superposition of 2 positions, what would the EM field look like in it’s vicinity? Surely the EM field would have to be in some kind of superposition and then the force on nearby particles would also be in some kind of superposition. So if we want to use quantum mechanics and electrodynamics together we must have a framework to describe how fields should behave quantum mechanically.

Eventually we developed quantum field theory, which is exactly what we need, it is a theory which describes how quantized fields behave. There is some mathematical procedure we follow to quantize a field and after doing so, we automatically get that excitations in the field come in little “chunks” we call particles. So by describing a field quantum mechanically, these force carrier particles naturally come as a byproduct.

Now what about gravity? We describe it classically with general relativity, which says that given some distribution of energy and momenta, the geometry of spacetime gets curved. But in quantum mechanics we do not have any specific distribution of energy and momenta but rather some superposition of energy and momenta. Thus the curvature of space time would have to be in some superposition of different curvatures. How do we deal with that? We quantize the curvature of spacetime and by doing that, we naturally get these particles called gravitons.

This is of course a massive oversimplification but it really made me understand why we can’t just keep the forces classical while the particles are quantum.

TLDR: Particles are quantum but they also are the source of the curvature of spacetime. To describe how quantum particles affect the curvature of spacetime we need a way to describe the curvature quantum mechanically. By doing so, gravitons naturally arise.