Illinois was (legally) taking their time sending in defensive subs when USC was trying to go fast. This was getting under Beamer's skin.
There is another situation. On the kickoff where the hootenanny happened, the SC receiver put his arms out straight. This is an almost university signal to tell the receiving team, "I'm not running this out, we'll take the touchback." This is an unofficial signal and not part of the fair catch rules. So the SC receiver did the arm thing, caught the ball, hesitated and then threw the ball across the endzone to the second receiver, who ran it to the 25 (which is where they were going to have the ball if they took the touchback). While there is no rule against this, it is in the family of a fake kneel down.
So, an Illinois player was injured a few plays later and SC had all the momentum. BB came out to check on the player. He turned to the SC sideline and extended his arms. THIS IS JUST MY GUESS. He either was saying, "Just so you know, we're subbing this guy out." Or, in reference to the kickoff, "Nice bleeping play you cheap bleepers."
Either way, Beamer totally lost control of himself and BB lived rent-free in Beamer's head for the rest of the game.