The splitting principle is as follows.

Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map $p^*: K^*(X) \to K^*(F(E))$ is injective and $p^*(E)$ splits as the sum of line bundles.

My question is, what is the idea/intuition behind the proof of the splitting principle?