How do electric fields accelerate particles?
Imagine we have a constant electric field:
If we have a particle with charge
Using Newtonian definition of acceleration,
So the particle has a constant acceleration in the same or opposite direction as the electric field (depending on the sign of the charge).
If you want maybe a more physical answer, the central question is something more like where does the energy come from? The particle begins at rest but then it accelerates without anything else being reduced. However, in order to keep the field static, we need to put energy into the system since we need to cancel out the particle's electric field itself.