Since you are a math guy, start here:
wiki
Basically, the presence of particles, or virtually anything, changes the metric of whatever manifold you are looking at. (Or it changes the Hamiltonian, whatever.) The effect is that the Levi-Civita connection (or whatever structure you are considering on your manifold) is changed, which means that geodesics (or your Lagrangian submanifold) get changed. This is the fundamental paradigm of general relativity.
So when a charged particle is present, the geodesics of nearby charged particles get changed.
Finally, in physics lingo, particles choose paths that are critical points for the "action." In other words, they try to minimize the "work" that they have to do, or at least try to find a path that is a local minimum (the mathematics is essentially the same as 1st semester calculus where one finds local minima via the derivative, etc. Look up the calculus of variations.)
Anyway, this is not really my area of expertise, so there may be technical difficulties in my exposition. But the general idea is that "fields" are simply attributes of the manifold (usually some sort of configuration space) that one must take into consideration in any physical problem. Strictly speaking, they are smooth functions from M to E, where M is your manifold, and E is some sort of bundle over your manifold.