Hey, I wrote a paper on this!

The short version is that there had always been some tensions between Scandinavians and the Sami. The sagas record that the Norse would forcibly extract taxes from them by entering the region and essentially extorting them into handing over furs. They were conceptually an embodiment of the "Other," often associated with unwholesome magic, shapeshifting and so forth; these motifs continued to hold currency up into the modern era. Tensions increased in the early modern era as Scandinavians moved north and began encountering the Sami with more regularity. Religious issues, in particular, influenced early state policy regarding the Sami, who were strongly encouraged to Christianize.

This all ramped up in the 19th century as Sweden and Norway looked to modernize and become true nation states in a romantic nationalist sense. Norway's policies of fornorsking, Norwegianization, stressed education in the Norwegian language for the Sami and kvens as necessary to integrate them into the Norwegian nation, and this goal was sometimes accomplished by removing children from their families to educate them in state-run schools. Sweden's policies were generally more relaxed but operated on the same principles. In the latter 19th century these efforts were linked to broader progressivism: social reformers who looked to help the marginalized in society identified the Sami as one group needing help, and the solution was more or less forcible integration into Scandinavian society. This took an even uglier turn with the birth of "race science," of which a number of Swedes were particularly strong proponents. The standard nonsense concerning racial traits, skull shapes and so forth came into play, and new developments in anthropology, biology and medicine were pressed into use to characterize the Sami as racially different and in need of saving or extinction.

Unfortunately medical sterilization did occur, as you note, but my reading suggested it wasn't as terribly common as reading about a "sterilization program" may suggest (though this hardly excuses the horror of this practice, of course.) Legally this occurred under laws influenced by a peculiarly progressive Scandinavian eugenics which were also sometimes used against the mentally ill or those deemed "antisocial." Fortunately Scandinavian "race science" fell from favor with the scientific establishment before the outbreak of WWII, and to such an extent that some particularly hard-line scientific racists felt compelled to move to Nazi Germany to continue their work. The vestiges of racial science lingered on through the 50s and 60s but the conceptual framework for this had largely fallen out of place, and the old institutions were largely quietly allowed to dissolve.

Finland was little bit different. Instead of proclaiming racial supremacy Finns promoted extreme equality, ie there was nothing special about the Sami. Samis were just another Finno-Ugric tribe, who needed to learn standard Finnish instead of their own shitty lingo. Methods to force this assimilation were however in many ways in par with the neighbouring Germanic nations.