The philosophical interest of the proof lies in its "bare hands" nature: there is no use of arbitrary axioms, just obvious truths about small finite structures. Yet the result can hardly be called a tautology. And the proof provides understanding of why the result must be true.

Take any point. Of the five lines from it, three must have the same colour, say filled. Consider the triangle formed from the three points at the end of those lines. If the three lines of the triangle are all dashed, there is a triangle of one colour. If one of the three lines is filled, it and the two lines back to the original point form a filled triangle. So in either case, there is a triangle of one colour.

More on the mathematics: A brief introduction to Ramsey theory

An introduction to mathematical proof from an Aristotelian point of view: Proof in Mathematics: An Introduction

A short article on Proof in Mathematics