How do fields work?

By fields, I am referring to gravitational, electric... Basically, how can two things interact at a distance? I know that we have mathematical descriptions of the phenomenon, but, physically speaking, I don't get it.

Physics is the discipline that describes nature using mathematical models , and assumptions that are axiomatic, taken from observations , called laws,principles, postulates. Axiomatic means that "it is what has been observed". The fields you describe are classical fields that arise in the mathematic fit of the data and are chosen because the mathematics is also predictive of new boundary condition solutions.

Fields arise because of the mathematical modeling of observations.

The fields you describe are classical fields.

Do particles, somehow, emit some kind of bullets, continually and in all directions that cause other particles to move, and, also somehow, one's "bullets" do not hit another's? ...

This is interesting because you are describing by words the complex mathematics used in quantum mechanics, where the classical fields can be explained by the exchange of virtual photons between interacting charged particles.

Here is how the interaction between two electrons is described in quantum field theory:

they exchange a "virtual" photon, another mathematical representation, carrying the quantum numbers, but not the mass of the photon.

The interesting thing is that this mathematical exchange of virtual photons can be shown to build up the classical electric field .

So in a pictorial mathematical sense, yes, a lot of virtual photons are playing ball, and the balls do not interact with each other because photon photon interactions are very improbable even for real photons, and more so for virtual ones.