I'm not sure if this is really an answer to your question, but I like to think about the splitting principle as the statement that if you want to check a formula for all bundles, it usually suffices to check it for sums of line bundles.

As Anatoly explained, this "principle" works because you can always pull back your bundle $E\to X$ so that it becomes a sum of line bundles, and moreover you can do so using a map $f: Y\to X$ that's injective on cohomology. So to check some (cohomological) formula involving $E$ in the ring $H^*(X)$, it's enough to check it in the larger cohomology ring $H^*(Y)$, and back in $Y$ you get to work with the sum of line bundles $f^* (E)$.

Often the real work will come in translating a formula that works for sums of line bundles into a formula that makes sense for arbitrary bundles. In the case of the Chern Character, for example, one has to introduce the Newton Polynomials for precisely this purpose.

I'm not sure if this is really an answer to your question, but I like to think about the splitting principle as the statement that if you want to check a formula for all bundles, it usually suffices to check it for sums of line bundles.

As Anatoly explained, this "principle" works because you can always pull back your bundle $E\to X$ so that it becomes a sum of line bundles, and moreover you can do so using a map $f: Y\to X$ that's injective on cohomology. So to check some (cohomological) formula involving $E$ in the ring $H^*(X)$, it's enough to check it in the larger cohomology ring $H^*(Y)$, and back in $Y$ you get to work with the sum of line bundles $f^* (E)$.

Often the real work will come in translating a formula that works for sums of line bundles into a formula that makes sense for arbitrary bundles. In the case of the Chern Character, for example, one has to introduce the Newton polynomials for precisely this purpose.