He seemed to be at least able to handle the hand injury, but once his mobility declined to the point where he was limping up and down the field the entire time and lost all his mobility, Utah launched their comeback (in case you missed it, Utah won the game 47-24 after Williams couldn't move).
Lincoln Riley talked about Williams' injury after the game. Turns out the injury was on the play where Williams heroically evaded the entire defense trying to tackle him for 59 rushing yards. That happened in the second drive of the game. Now it's clear why Williams didn't have any heroic runs after the second drive of the game.
The injury was a popped hamstring, according to Riley. Riley explained that despite him not even being at 50%, Williams would not let Riley take him out:
USC Football QB Caleb Williams truly did put up an incredibly gutsy performance that should still win him the Heisman Trophy.
USC Football QB Caleb Williams, on one leg, still completed 28 of 41 passes and dropped three end zone dimes while only throwing one pick. He threw for 363 passing yards, so it was an efficient day, and especially so when considering the defense that Williams was playing against.
It's not easy to throw on Clark Phillips III and the Utes' D who entered the game top 19 in the country in scoring defense (20.1 points allowed per game).
With Williams' best offensive lineman out (LG Andrew Vorhees), and then his second-best offensive lineman out when SC was driving on Utah's side of the field and about to make it a one score game with 7:37 left (C Brett Neilon), the pocket continued to collapse. It already had been collapsing due to Vorhees' injury. Williams didn't have the mobility on one leg to get out of that pressure.
The injuries were piling up on this team. Remember that the best pass-blocker in USC's backfield in Travis Dye was obviously out for the game himself. He had no protection against a very good defense, yet still put up quality numbers. It's hard not to respect what Williams did, and how he toughed it out. He will likely be rewarded with the Heisman Trophy next week.