I know why they did it: because they were already fighting the USSR, a marriage of convenience.
Bizarrely, I recently learned the Finnish front had both Nazi SS units and a field synagogue for Finnish Jews, fighting on the same side. Even weirder, the Nazi SS unit in Northern Finland was formed from former concentration camp guards.
3 Finnish Jews were even offered the Iron Cross by the German command for saving many German lives (All 3 refused).
http://www.jewishquarterly.org/issuearchive/article8d14.html?articleid=194
Captain Solomon Klass saved a German company that had been surrounded by Soviet forces. Two days later, German officers came to offer him the Iron Cross. He refused to stand up and told them contemptuously that he was Jewish and did not want their medal. The officers repeated their ‘Heil Hitler’ salute and left.
Finland didn't see itself as on the Axis side per se. It saw itself as embroiled in an existential struggle, lacking allies, and with nowhere else to turn. Keep in mind the chief reason Finland wasn't part of the Soviet Union was that it declared independence in 1917 and the "white" forces won. In an era when the Soviets were absorbing countries like Poland, Finland's position was precarious at best. The Allies supported its independence but weren't prepared to take more than token gestures to ensure it.
A complicating factor in this narrative is the splitting of the conflict into the Winter War (essentially defensive) and Continuation War (in which Finland tried to annex Soviet territory). To understand Finland's actions in the latter affair we must look at Karelia, a strip of land that was at the time divided between Russia and Finland. Many Finns viewed Karelia as the spiritual birthplace of their country; you can find Ervasti talking about how "for many years I have regarded Russian Karelia as part of our mother country of Finland" long before WW2. So there was some irredentism as well. After the war, ethnically Finnish areas including Vyborg became part of Russia, which is why it's an ongoing issue for the Finnish side.
Source: Finland in World War II: History, Memory, Interpretations edited by Tiina Kinnunen and Ville Kivimäki