The humor in this skit derives mainly from the juxtaposition of his normal persona with outbursts from his repressed id, culminating in a scene that shows how the latter has overtaken the former.
My father left my mother for the love of a (POONTANG) -nother.
This joke insinuates that his father didn't really leave for love, but instead for tawdry, midlife-crisis sex. It's also similar to the common practice of punctuating the word another with obscenity (e.g., a-fucking-nother), called expletive infixation.
In primary school, I had trouble making (ASHTRAYS) friends.
This outburst refers to the formerly common practice of making clay ashtrays in grade school art classes. They're generally awful – and it's a terrible idea – and most of the people in the audience have probably done it. It makes for an absurdly innocent counterpoint to the other jokes, and it subverts the fixed phrase “making friends.”
And while he lay there bleeding, I used his feet to kick him in the head.
The joke here is partly in the sheer, grotesque absurdity of kicking a man (while he's down) with his own severed feet, partly in the irony that integrating his repressed feelings has simply made him homicidally vengeful toward the person who told him to do it, and partly in allusion to the childish game “Why are you hitting yourself?”