You keep your trap shut and I'll keep your trap open

Solution 1:

This is a saying among drug dealers.

Keep your trap shut:

Here, trap means mouth. So it means to keep your mouth closed no matter what the police, or anybody for that matter, say.

Keep your trap open:

A trap is a place (usually an abandoned house) where drug dealers make money and I'm talking money not just a few dollars I mean thousands and thousands of dollars 24x7. And so it means "I'll keep your business running".

References: Urban Dictionary 1 and 2

Solution 2:

"Keep your trap shut" is a colloquial expression meaning "don't talk". In this case the "trap" in question is the client's mouth. This is likely an invocation of lawyers' frequent advice for their clients never to speak to law enforcement officers without a lawyer present, since such loose talk is admissible in any subsequent court case and is almost exclusively detrimental to the defense.

"I'll keep your trap open" is not a common saying, but is rather a play on the other saying. It is likely supposed to mean that the lawyer will keep his client out of prison. In this case "trap" means prison.

Putting it all together, then, the lawyer is advising his potential clients not to speak to police until hiring him, thus maximizing his ability to prevent a lengthy jail sentence.