What does 'laugh' mean in this sentence?
"Remember that we could laugh you out of politics."
In this case, "laugh you out" is an uncommon (but recognisable) idiom for ridiculing you.
Effectively, this is a threat by the speaker to you, that they could make people laugh at you so much, that your credibility would be damaged and you would feel too embarassed to continue in your political career.
As mentioned in comments by @FumbleFingers, its use is more common in the form, "it was laughed out of court". This describes where the case made by the plaintiff or defendant were deemed so ridiculous that the judgement was swiftly delivered – possibly without ever going to a formal trial.