Per Raymond E. Zickel (Soviet Union: A County Study, 1991), this can largely be attributed to the growing General Assembly. Most of this growth came from newly formed States and ex-Colonies. These new admissions were largely anti-Imperialist and, as a side effect, anti-American. For it's part, the Soviet Union continued this rhetoric, allegedly while using the UN to foster these new nations economic development. Ideas such as New International Economical Order (NIEO) were used politically to continue this trend, especially among the Southeast Asian and South American states.

Previous to all of this, the United States and Western states were the dominating power among the United Nations, primarily due to early entrants of European nations aligned with the West/NATO and the over-representation of the West among the Security Council. If you review the Security Council Official Records (the 186th meeting, for example), the early years of the USSR's vetos were largely towards countering, what they perceived as, growing Western influence. This is largely evident in their delay ratifying the Austrian treaty and using that precept to exclude membership.