> However, I don't entirely grasp the physical meaning of conservative
> fields, I thought understanding where the label came from might help
> me understand what it is that we're describing.

It may be helpful to think in terms of the the conservative forces associated with conservative fields, like the gravitational and electromagnetic forces.  A conservative force has the property that the total work done by the force in moving a particle between two points is independent of the path taken between the points. For example, the work done in moving a mass between two points in the gravitational field does depend on the path taken. 

The term "conservative" refers to the fact that the overall mechanical energy (kinetic + potential) of a particle is conserved. An example where mechanical energy is not the conserved is when kinetic friction forces are involved.

Hope this helps.