It's the only logical reason as to why he would leave, right? He's in no jeopardy to lose his job at ND; if things break right, they could be in the final 4 again this season (although a few things would have to happen for that to occur). He has them consistently good, he's gotten to a title game, he sends kids to the league and I don't believe he's had any kind of significant scandal. Financially he's more than fine and could have used this likely to get more money from ND.
The only reason has to be that he sees LSU as giving him a better chance of getting a title because of how the SEC is elevated compared to other leagues, and because LSU is still one of the better jobs in that league. It is probably also easier for him to land a particular type of recruit at LSU compared to ND, whether because of academics (I believe ND still is more selective there than other high major football powers) or geography -- getting kids from Georgia/Florida hotbeds, or in state, likely an easier sell at LSU.
Interestingly, I think Riley went to USC for a similar reason, albeit going away from the SEC. His contract details, if as reported, are insane -- more and significant $, real estate stuff, apparently use of the private jet for his family whenever he wants. So the perks are insane. But I also think the "big picture" of where he can take Oklahoma compared to USC played a role in it. The road to the championship/football play gets significantly harder for OK with the move to the SEC. In the Big 12, they were the true power team during his tenure (this year aside), especially with the drop off of Texas. He could schedule a big opponent OOC and if he won the conference with perhaps 1 loss, he had a good chance of getting into the football playoff. Now OK will be competing in a much deeper conference and with the big boys of Bama and Georgia. I think he sees USC as a place he can recruit at a high-level, replace Oregon at the top of the Pac 12, and have a legit shot of being in the playoff each year with a much easier road.