"What's his beef?" is the equivalent of "what's his complaint?" or "what is making him angry?" According to my quick internet search, this expression dates to the early 1900s, but the details of its origin are lost to history.

The basics of the meaning "what's his beef" have already been answered, but it's one of those vernacular expressions that has a range of meanings. Here, though, the range is less in meaning and more in strength of intent, as the "beef" in question can mean cause of his being sullen (i.e. can be very petty) through to a full on vendetta (usually expanded to clarify what the "beef" is with i.e. "what's his beef with [insert someone or something]?", which means that the person asking the question is aware of "his" vendetta or expression of loathing/contempt for said someone or something and wonders why).

'What's his beef' means 'what's his problem'.