What is actually a 'Field' in Physics? How can something affect something at a distance?

Every text just describe fields mathematically and as a 'vector field' in which it is said a particle gives rise to a field because each point in space around it becomes associated with with a force vector. But it never explains how does a particle generate a field at the first instance, or what a field really is or how does a field work - how is a particle able to affect another one at a distance?