Our best theories, like the aforementioned theory of evolution, the Big Bang theory, and Einstein’s General Relativity, cover all of these bases. They have an underlying quantitative framework, enabling us to predict what will happen under a variety of situations, and to then go out and test those predictions empirically. So far, these theories have demonstrated themselves to be eminently valid. Where their predictions can be described by mathematical expressions, we can tell not only what should happen, but by how much. For these theories in particular, among many others, measurements and observations that have been performed to test these theories have been supremely successful.
But as validating as that is — and as powerful as it is to falsify alternatives — it’s completely impossible to prove anything in science.