For the American point of view, there are three major points. First, the US provided huge amounts of materiel to the USSR and to Britain. These weapons and other equipment allowed the Allies to win the war. Second, the US (and Britain), and not the USSR, engaged in the strategic bombing that helped cripple Germany. Finally, the war could only end when the Western Allies invaded Europe.
For the Soviet point of view, the main support is the number of German troops the Soviets killed and took prisoner and the huge burden the Eastern Front put on German resources. The Eastern Front was where the majority of German soldiers died. The Soviets also lost tremendous numbers of troops and civilians. The sacrifice by the Soviets, in this view, bled the Germans and weakened them enough so that they fell when pressured from the West.
I am not at all sure that most Americans believe, as you suggest, that the U.S. won the war with a little help from the Soviets. The help from the Soviets was tremendous. If the Russians want to believe that they won the war with a little help from the U.S., they could make a pretty good case for their belief. The important thing was to beat Germany--and it took everything the Soviets, the U.S. and the British could throw at them to make them finally surrender. The pictures we see of their bombed-out cities are evidence of the Anglo-American contribution to the victory. The war didn't end with the German surrender. Japan still had to be defeated, and that half of the war was won by the U.S. virtually alone.