If two particles are close enough for their charge to affect each other, they'll start to move if nothing is blocking them and that means it will accelerate. We know that scceleration is the change in velocity (or speed) divided by the time required. If the electron wasn't moving at some point in the past, but is moving now, then its velocity changed. That means it had an acceleration.

From Newton's Second Law, we know that things accelerate when there is a net force on them. Therefore, for an electron to accelerate, there must be a net force on it. The question really comes down to "what will exert a force on a stationary electron?".

Basically, an electron moves because it started to move, and it started to move because of the presence of another charged object.

This info was found in several links but they're all based on Coulomb's Law and since motion/acceleration are involved, Newton's Laws of Motion.