Why does an indicator need to be a weak acid or base?

Because the way it works is by a change in structure due to pH, usually protonation or deprotonation. If it weren't a weak acid or base, moderate changes in pH (which is what we use indicators to detect) would have little to no effect on its structure.

Let's call our indicator in its acid form HIn, and its base form In-. Above a certain pH, the base will lose the proton and become predominantly the In- form, which is one color. Below a certain pH, the base will gain a proton and become predominantly the HIn form, which is another color.

Think of it like a buffer solution of a weak conjugate acid / conjugate base pair - if you exceed the capacity of the buffer, it will be predominantly be one or the other. (We usually use very small quantities of an indicator, making it easy to "exceed the capacity of the buffer".) That's how indicators work as well.