In physics, if we see flow, it is because there is a gradient, or difference in the amount of something in one region of space versus another region. For example, if we consider water flows down a waterfall, we will realize that there is a gravitational potential difference between the top of the waterfall and the bottom. Namely, there is higher gravitational potential at the top, and lower gravitational potential at the bottom. In the case of the waterfall, water flows from high gravitational potential to low gravitational potential. Something similar happens for electrical charges. A voltage is a difference in electrical potentials in one region of space as compared with another region of space. If there is a difference, electrical charge will flow from regions of high electrostatic potential to low electrostatic potential. I hope this helps!

If there is no potential difference, why would the charge flow? You may start with this question. When you apply the potential difference, you are supplying the external force that can drive the charges. Another similar example is the water fall. If there is no external force, and then the water would stay at the same place. Since water (or everything) is subject to gravity, so whenever there is a chance, the water would flow from a high position to a low ground.

Electrons, like any matter, will not move unless acted upon by a force. This is Newton's first law of motion.

Electric charges generate a force between them: like charges repel each-other, while opposite charges attract. Electric potential difference is a measure of the strength of this force, divided by the amount of electric charge being acted upon. As such electric potential differences are needed to make electrons move, i.e. create electricity, because the potential represents the force needed to get past Newtons first law.

If neither the positive charges nor the negative charges are being 'pulled' in some direction, they don't go anywhere. There needs to be somewhere that the positive charges, or the negative charges, 'want' to go, in order for current to flow. That's the potential difference - the force pulling one of the charges in one direction.

Think of electric potential energy just like you think of gravitational energy. If you hold an object high above your head, it has a lot more potential energy than it does if you were to hold it just above the ground. If you look at the formula for electric potential (k q / r), we can see that r, or the distance between the charges, is just like the distance between the object and the ground in our gravitational potential energy comparison. Objects in space will always move from a place of higher potential to a place of lower potential, like an apple falling from a tree toward the ground, therefore we can see that the difference in potential is what makes electrical particles move!