In poker, at the end of each hand, the winner "lays their cards out on the table" to show they've won. Since part of the game of poker is to try to disguise your hand from the other players, when the winning hand is laid on the table, this reveals facts that previously have been hidden or even deliberately lied about (through bluffing) by the players.
This leads to a common idiom, "lay your cards on the table" to mean showing your actual capabilities. In a negotiation, this idiom can mean revealing your actual wants and what you're willing to give in return.
The idiom is also used in a more general form, "lay [something] on the table" and with a broader meaning, to reveal things that are hidden or simply to speak plainly.
In your examples, there are people who might not normally reveal their true beliefs, and the writers are using the idiom of putting them "out on the table" to mean expressing those beliefs openly.
"On the Table" also means for something to be subject to discussion or consideration by a group. In the written example you provided, the characters views about premarital sex were "on the table". I would infer this is not a subject matter they speak often about.
The "plates" being referred to here are likely license plates:
"Run your plates" is a set phrase, mostly used by police.
It generally describes a process of noting the license plate number, looking it up on a computer, and using the data there to figure out the name and address of the person who owns the car.
I've not seen this film, so I'm not familiar with this scene but I'm pretty sure this is what Cooper means.
Best Answer
The highlighted passage can thus be read as...
The second citation just looks confusing because there are three different pronouns all refering to the same person (you when "they" are addressing the speaker, he when one of "them" is speaking to someone else, and I when the speaker refers to himself).