As soon as the war began to turn sour for their German co-belligerents in 1943, Finland began to seek a graceful exit from the conflict. Secret negotiations were conducted by the Finnish government with the Allies in 1943 and 1944, but no conclusive agreements were ever reached. Meanwhile, the USSR began a bombing campaign against Helsinki in early 1944, hoping to force Finland to surrender. These raids were ultimately a failure both in tactical and strategic terms; very little damage was done and Finland felt no urge to surrender, especially as the Soviets offered more and more unfavorable terms. The Red Army broke the Siege of Leningrad in January 1944, and drove the German forces back towards Narva. This prompted Finnish Field Marshal Carl Gustaf Emil Mannerheim to tell German High Command that Finland would surrender if Axis forces were driven from Estonia, as the Soviets would be able to conduct further air attacks over the Gulf of Finland from recaptured bases there. In lieu of this, the Red Army launched an offensive against Finland in June 1944 that drove their forces back towards the city of Vyborg. While the Finns could not hold their ground, they did manage heavy losses on the attacking Soviet forces, who eventually ground to a halt. The offensive also drained the Finnish Army of any further resources to conduct the war.

However, President Risto Ryti had made an agreement with German Foreign Minister Joachim von Ribbentrop that Finland would not seek a separate peace. This was taken only as a personal understanding. Seeing the writing on the wall, Ryti resigned on 1 August and Mannerheim took his place as President of Finland. Mannerheim cancelled Finland and Germany's agreements and sought peace with the USSR and Britain on 17 August, and received said peace terms from the Soviets on 29 August. These terms were (after further negotiations in September): Finland must demobilize its forces; remove any German forces from within its territory; return its borders to those of the 1940 Moscow Armistice that ended the Winter War, plus cede the Petsamo region to the USSR; pay $300 million (app. $4.3 billion today) in war reparations over six years; legalize the Communist Party of Finland and ban any parties the USSR deemed fascist; and try any individuals the Soviets considered responsible for the war.

As for why Stalin did not ask for more, this is fairly debated. First, the Soviet Union never held much of Finnish territory; it would have been seen as blatant antagonism by the western Allies for the Soviet Union to try to force a regime change in the Finnish government or take the entirety of Finland. Pressing for further concessions may have led Finland to leave the negotiating table and force the Soviets to take further losses. It is very likely Stalin did not want to continue needing to redirect his forces to hold the "Finnish sideshow." At the Tehran Conference in 1943, Stalin said his only objectives in Finland were territorial concessions, reparations, and military bases. Finnish military leadership, having learned Soviet military forces in the Leningrad region were to be transferred to the reserves, also knew the Soviet Union was no longer interested in occupying Finland by 1944. The Soviet Union wanted to focus on beating Germany, not conquering Finland.

For my sources, I will suggest:

Raunio, Ari; Kilin, Juri (2008). Jatkosodan torjuntataisteluja 1942–44 [Defensive Battles of the Continuation War 1942–44] (in Finnish). Keuruu: Otavan Kirjapaino Oy.

Vehviläinen, Olli (2002). Finland in the Second World War: Between Germany and Russia. New York: Palgrave.

Ylikangas, Heikki (2009). Yhden miehen jatkosota [One Man's Continuation War] (in Finnish). Otava.