What is the Pumping Lemma in Layman’s terms?

I saw this question, and was curious as to what the pumping lemma was (Wikipedia didn’t help much).

I understand that it’s basically a theoretical proof that must be true in order for a language to be in a certain class, but beyond that I don’t really get it.

Anyone care to try to explain it at a fairly granular level in a way understandable by non mathematicians/comp sci doctorates?