The confusion is with the word "about" - it is a comparitive: I feel about as hungry as an elephant I was about as happy as a bird in a tree So in this case he looked about as stupid as I felt.... the narrative is not written well

The grammar is correct, although the writing could be clearer, so you really have to know English well to sort it out. First of all, no one is looking about. The narrator’s dad was looking stupid — about (i.e. approximately) as stupid as the narrator felt (himself to be). Second, the narrator was standing by himself. The reflexive form (-self) refers to the subject. The narrator was standing by himself. No one else can stand by “myself”; everyone else can stand by “me”, but only I can stand by myself.

my dad was over with Mrs. Baker looking about as stupid as I felt - this phrase describes the dad. If this confuses you, try removing "about". standing by myself talking to no one. - this phrase describes the narrator/the son. He felt stupid because he wasn't interacting with anyone, he was just standing there. "About" in this context can be replaced by "around". These words are used when estimating. My brother is about/around as tall as my dad.

Her dad was Looking about, standing by myself is the story teller. Everything is correct.

The sentence is fine but is not very formal. He is saying that his dad looked stupid whilst talking to Mrs Baker and he felt stupid because he was on his own and didn't have anybody to talk to. He is comparing how stupid he felt (social awkwardness etc.) with how stupid his dad looks.