Perhaps the most famous and certainly one of the oldest visual explanations in mathematics is this visual proof of the Pythagorean theorem. This proof is unusual in its brevity and its complete appropriateness to the problem.
Some diagrams are more useful than others.
Edward R. Tufte has written three books analyzing the effectiveness of figures and diagrams in communicating ideas. Casselman focusses on the applications of Tufte's ideas to explanations in mathematics, and distills from the book a set of rules designed to make graphics contribute most effectively to the communication of mathematics. Replace coded labels in the figure by direct ones. Produce emphasis by using the smallest possible effective distinctions. Eliminate all unnecessary parts of a figure.