What do they mean by "an intellectual" in this context? Seems to be used in the jokes where they say something witty as a punchline, but it feels like mentionin

What do they mean by "an intellectual" in this context? Seems to be used in the jokes where they say something witty as a punchline, but it feels like mentioning it explicitly doesn't add anything? Or am I missing some nuance? On re-read, it seems that it might be even ironic - a supposed intellectual saying something dumb?