It's the convention that we define the direction of the electric field by the direction that a positive test charge would be pushed. Because a positive test charge would be repelled by positive source charges, we draw field lines and pointing away from positive source charges. Because a positive test charge would be attracted towards negative source charges, we draw the field lines as pointing towards them.
Both positive and negative charges will spontaneously move towards the lowest possible potential energy state. Imagine an electric field of two parallel plates, one positively charged and one negatively charged. If we were to introduce a positive test charge into the middle between them, it would naturally want to run towards the negative plate. What if we physically stopped it from doing so, and dragged it towards the positive plate? Well, it'd be very repelled by the positive plate, and it would be very uncomfortable. The closer we drag it to the positive plate, the greater its 'potential energy.' The more we drag it, the faster it's going to run away if we let go. Imagine letting go of the positive TC when it's close to the positive plate - all that potential energy will quickly be converted into kinetic energy!
What if we were using a negative test charge? Well, instead of dragging it, let's just leave it hanging out somewhere near by a positive plate. As you can guess, it's going to begin picking up speed and accelerating faster and faster towards the positive plate. Well, it's gaining all this kinetic energy now, so it must be losing potential energy. So we see that as the negative charge gets closer to the positive plate, its potential energy becomes more negative. This also means that the potential energy of a negative charge is highest (0) when it is infinitely far away from the positive plate.
The electric field describes the nature of a region of space. It tells us "if there were a positive charge here right now," it would do this. The potential energy tells us the same thing, but instead of telling us about force, it tells us how much energy a positive charge would have in that spot. So because positive charges have high potential near positive source charges, we say that the positive plate is the region of high potential. What happens when a positive charge is infinitely far away from a positive plate? We say that it's potential energy is 0. As it gets closer, the potential gets more positive.
What if a negative charge is infinitely far away from a positive plate? Its potential energy is also 0. But unlike a positive charge, as we move closer it gets more and more negative.
In summary:
We define the electric field as the sum of all the forces acting on a positive charge.
We define the potential energy of a region of the electric field as being the energy held by a positive charge in that region.
Positive test charges will move from high potential to low potential because this is how we define the electric field.
Negative test charges will move from low potential to high potential because the potential is defined as for a positive charge, so it makes sense that negative charges will move in the opposite direction.
If we instead decided to say that "regions of high potential are where negative test charges will experience the highest potential energy" then the negative plate would be the one with the high potential, and the negative charges are the charges that would go high to low, and the positive charges would go low to high.
Hope that helps.
