I have seen this definition almost everywhere, but I don't know the underlying proof behind this definition.
These definitions are from Wikipedia
In theoretical computer science and formal language theory, a regular language is a formal language that can be expressed using a regular expression...
Alternatively, a regular language can be defined as a language recognized by a finite automaton.
You can define a regular language as a set of strings accepted by some FA. In this case you don't have to prove anything since a regular language is recognized by a finite automaton by definition.
But if you define a regular language as a formal language that can be expressed using a regular expression then you can prove that it is accepted by some FA using Thompson construction, i.e. prove the equivalence of regular expressions and finite automata.