What happens in the primary coil
Electrons whose motion is forced into curved paths align their magnetic dipoles (and yes, each electron is also a tiny magnet). The core of the transformer directs this common magnetic field through the secondary coil.
What happens in the secondary coil
In the case that the magnetic field changes, the electrons in the secondary coil are aligned by this field and experience a shift (Lorentz force, Hall effect). Since for all electrons the magnetic dipole and the spin are related, all electrons are shifted in the same direction (an empirical fact that allows us to use inductive processes in everyday life).
Does a higher voltage in the secondary coil mean that all electrons in the secondary coil are taken to a higher potential?
In the secondary coil, the changing magnetic field induces electron displacement along the entire wire. If the length of the wire is short in relation to the primary coil, more electrons can be shifted per mm and this leads to a higher current. Since the power of the (ideal) transformer must be the same on both sides, the voltage on the second coil must be lower.
If on the other hand the secondary coil has more turns, fewer electrons per unit length can be displaced; the current is lower and the voltage higher.
And why is the current (flow of electrons) low when that happens?
The power on both sides of the transformer must be the same (without taking power losses into account). The power is the product of current and voltage P = U*I. If more electrons are affected in the secondary coil (higher current), the voltage is lower and vis-a-vis.