A mathematical logic is a formal system of formulas (axioms) and rules for deriving other formulas, called theorems.
A proof is a derivation of a theorem. To see a concrete proof tree, click here.
Why should you care? The neat thing about Theorems is that they are ``true.'' More precisely, if all the axioms are valid and the rules are validity preserving, then anything derived from the axioms via the rules is valid.
So, if you want to determine if some formula is true, prove it.