Why is being "reduced" equivalent to having a greater potential energy?

Answer 1

When people say 'reduced' they mean a reduction in charge. A proton has a positive charge (but it is integral to the nucleus, so it isn't going anywhere) and an electron has a negative charge. Since electrons are highly mobile, a reduction in charge almost always means that you've tacked on an electron. The reduced compound has more potential energy because it can pass the electron on to another compound.

It's kind of counter-intuitive that a negative charge would mean the presence of an electron. Think of it as a gas gauge in reverse that reads 'E' for full, and 'F' for empty.