believe me

Michael Quinion has written about this phrase here.

I think this is a case of extending the command mode
in this case, the verb is what must be followed so it comes first.

(you) picture a captain commanding his troops
"Company Halt!"
{company)"About face!"
(company)"March!"

The subject "Company" is dropped from the commands to get right to the action

I think in the case of "believe you me",
the "you" was reinserted after the verb to make sure the sentence had a subject on paper.
In essence, the writers were trying to be grammatically correct by reinserting the "you".