Here is a differential geometric point of view, which of course works only over $\mathbb C$.
Since $L\to X$ is ample, then it carries a smooth hermitian metric $h$ of positive definite Chern curvature $i\Theta(L)>0$.
Now let $f\colon Y\to X$ be any finite morphism. Then $f^*L$ inherits a smooth hermitian metric which is just the pull-back oh $h$ and its Chern curvature is given by the pull-back
$i f^*\Theta(L)$. Since $f$ is finite, then its differential is (more or less) injective, so that
$i f^*\Theta(L)$ is positive definite, too.
But then, by Kodaira's projectivity criterion, $f^*L$ is ample.