The Thalmor were the aristocratic, arcane and intellectual elite of Summerset Isle. An class of stuffed shirt magi who had steadily declined in influence and power since Summerset was absorbed by The Empire in the second era.
In the turmoil of the oblivion invasion at the end of the third era, they rose up to push the daedra invasion back beyond the gates of oblivion and thus save their people. Sort of. Their victory made them enormously popular with the common folk of Summerset Isle.This popularity came at the expense of imperial pride, of which the altmer were already seriously lacking.
The Thalmor used their new-found hero-status to rabble rouse against The Empire. Claiming that The Empire was an impotent paper tiger that refused to come to their aid in The Oblivion Crisis. This, of course, ignores the fact that no province was hit harder in the oblivion invasion than The Imperial Province of Cyrodiil, with Mehrunes Dagon himself stepping all over Capitol City. It took a literal act of divine intervention to save Cyrodiil (and all of Tamriel) as the avatar of akatosh defeated Dagon and ended the invasion and the third era.
In the early fourth era the Thalmor lead Summerset Isle seceded from The Empire and took up its ancient elven name of Alinor. Agents of the Thalmor within the Valenwood societal elite staged a bloodless (sorta) coup and handed neighboring Valenwood over to Alinor. The ancient Aldmeri Dominion (The Elf-Home Dominion) was born again.
Then you have some shenanigans with the khajiit and the missing moons that resulted in
Elsweyr throwing in with the Dominion. The great war, the white-gold concordant that ended it and finally the signing of the second treaty of stros m'kai. Then we have the civil war within the empire, and that's where we stand.
They started as a small group who used societal malcontents in the wake of a major war to seize and cement power over their people. They believe in the superiority of elven kind, though not all elves are equal. The Altmer are the apex of elven blood, the Thalmor are the greatest amongst them. The bosmer are lesser elves, fit to serve them, but nothing more. The dunmer are heretics to be enslaved or destroyed. The Orsimer are perversions of the elven form, monsters to be slain.
They believe that all of Tamriel is theirs by right of heritage. They believe that the races of man should be destroyed, enslaved, or pushed back to Atmora. They do not
truly care about the worship of Talos, as their ultimate end-game is to destroy humanity anyway. Banning it was simply a ruse. A ploy. A political gambit to sew unrest within The Empire. It has worked perfectly. The Empire fights itself attempting to enforce The White Gold Concordant, The Dominion grows strong.