Fields in QFT have a particle content too, so the field is not replacing particles, they are just described in a different way as the possible modes of vibration of the field. Now fields originate in classical physics, most notably with the work of Maxwell. It was then realised that a field solves many problems, like instantaneous interactions at a distance, and provide a reservoir of energy and momentum that gets transmitted through waves. This is where the particle interpretation settles in, as the interaction is carried by a quantum intrinsically associated to the field.
Another problem that is solved by fields is that when relativistic effects are considered, the notion of particle is not well-defined. Suppose you want to describe a particle that is decaying. You start with a system described by a particle in a certain state and you end up with a different final state with a different particle content. Hence it is difficult to attach a state to a particle that after a while disappears. What has not changed is the total energy and momentum of the system, and this can be explained as different fields that communicate their energy and momentum among themselves through the way the interact with each other.