Since the term is slang, and pretty modern, it's unlikely that there is anything written about the etymology of "beef".
Based on the limited information out there, it seems to have arisen from gang, street, or rap cultures in the United States. Most commonly it is used as a term for a fight, to start a fight, to have a cause for a fight, etc. As in, "don't start a beef with me because I will win."
There seems to be real roots in the underground rap/hip hop cultures, where rappers or artists would use "beef" to refer to the rap battles that they would have.
If I had to venture a guess, I would say that the term arose from the resemblance of your face after a street fight and ground beef, bloody and torn up.
Source: Pure speculation.