Why do unlike charges have greatest potential when they are further apart?

It's potential as in potential energy. As with gravitational potential energy, the higher an object is from the earth's surface, the more gravitational potential energy it has. In contrast, like charges repel, so the closer they are, the more potential energy they have.

Unlike charges attract so they have greatest potential when they are farthest apart. Think of potential as the capability of doing work. In principle you could put a brick in front of each charge and the charges would do work moving the bricks as they approached each other. If the charges started near together, there would be less potential for doing work.
Alternatively think about holding one charge in each hand and pulling them apart against the Coulomb force. You would have to do work which would be stored as electrical potential energy. The farther apart the more work.