Confusion over common criterion for a line bundle on a scheme to be trivial?

Question

Suppose you have a line bundle $L\to X$ on a scheme $X$. I'll denote by $\sigma\colon X\to L$ to be the zero section. Why is it that this bundle is trivial iff there is another section $s\colon X\to L$ such that $s(X)$ and $\sigma(X)$ are disjoint?