Well, first of course, he's obviously living there currently. Self-preservation would motivate anyone to try to prevent a hungry dragon from devouring the world they live on.
But to go more in-depth... Elsweyr is no more. The Altmer have assassinated the Mane (basically the Khajiit's Pope, to oversimplify things) and made a coup which has split the country into two separate and feuding entities, Anequina and Pellitine. They've also managed to pull the wool over most Khajiits' eyes so they are seen as the good guys by the majority of the population, despite having taken control of their land through shenanigans and lies.
Since Cyrodiil submitted to the Thalmor's diktat after the armistice, the only places that still fight the tyranny of the high elves are Hammerfell and Skyrim.
Your Khajiit might simply be an opponent to the Thalmor's rule of terror, forced into exile, and trying to fight back against his enemies where he can. He might even dream of gathering an army of angry Nords, join up with the Redguards, and cleanse continental Tamriel from the Thalmorian corruption.