Learn English – What does “This being….” mean here

"Figures" in this case means "to come to a conclusion". It essentially means that the person has used the premises that they have on hand to arrive at an understanding of the facts.

However, the phrase "figures you" is an actual expression that has a different meaning. If someone "figures you", it means they have decided you probably belong to some sort of group. For instance, "I figured you for a Yankees fan, not a Mets fan!", would mean the person thought that you were a fan of the New York Yankees, rather than the New York Mets.

In either case, "figure you" means they have come to some conclusion about you or your actions, but the case given is not the same as the expression "figure you".

The song uses this phrasing for the purposes of rhythm and meter. Dropping articles before words is commonplace in lyrics, poems and other creative writing.

So, to strictly answer you question: Yes it has meaning. It means the same thing as, "What does the fox say?" But no, it isn't grammatically correct in the sense that you would never use it in a formal context.