This is from David Foster Wallace’s handout on five common usage mistakes, which he gave to students taking his Fall 2002 section of English 183A (an advanced fiction writing class) at Pomona College:
For a compound sentence to require a comma plus a conjunction, both its constituent clauses must be independent. An independent clause (a) has both a subject and a main verb, and (b) expresses a complete thought. In a sentence like “He ate all the food, and went back for more,” you don’t need both the comma and the and because the second clause isn’t independent.
For a guy like me—that is, someone who writes and edits by instinct more than anything else—this is eye-opening. I mean, I could’ve told you that the comma doesn’t belong there, but for the life of me I wouldn’t have been able to explain why. Nor would I have had the slightest clue as to how to find out.
You can see the original handout here.