Why does he use "trap" here? I don't see "trap" is used in this way much.
Native speaker’s answer
Rebecca
"Trap" here is used to indicate that someone's mouth is "stuck" in grease, or else, "full" of grease. So a "grease trap in your mouth" here is just another way to say that someone's mouth is full of grease. "Grease trap", however, is not a general expression.