What does resolution really mean in a Large Eddy Simulation (LES)?

Does increasing the resolution have a strong effect on simulation predictions?

It depends on what you are trying to study. Many bulk flow and mixing processes in agitated vessels can be reproduced using only a few million grid points. Such simulations can provide very good predictions of system-level blend time, power draw, impeller pumping, etc. If you are trying to get more detailed insights into the local turbulence mechanics and micro timescales, you are going to want more resolution. I can send over some journal papers that discuss this point.

Mechanistically speaking, within the context of LES, the fineness of the grid spacing informs how much of the turbulent energy spectrum is resolved explicitly by the simulation. That is, the grid spacing informs the minimum eddy length scale that is resolved explicitly by the solver. Sub-grid motions are still incorporated into the fluid mechanics via the eddy viscosity, but their effects are not resolved explicitly. Sometimes this matters, other times it does not.