Here is what happened.
One set of officials called a forward pass, and blew the play dead after the ball hit the ground.
A second set of officials called a fumble, and wanted to allow the play to continue. The LSU player recovered the ball 1 yard short of the first down, and was not able to advance the ball because the play was blown dead. Therefore, the officials ruled that the result of the play was an inadvertent whistle, and gave LSU the option to replay the down.
HOWEVER! A forward fumble recovered by the offense can not be advanced. The play was over at the spot that the LSU player recovered the fumble. The inadvertent whistle was completely without consequence. It should have been 4th and 1 LSU.
Unfortunately, the officials, who are paid almost $100k a year, don't even know the rulebook well enough to get this call right. There was no situation in that play that would have resulted in an LSU 1st down. The officiating crew is either crooked or incompetent. It is one thing to miss a close call. It is a totally other issue when you don't even know the rules of the game well enough to make the right call.