What was the major push for the Jews to leave Russia?

Answer 1

Jews were largely forbidden from leaving the Soviet Union. The movement to allow Jewish emigration from the USSR was active in America and elsewhere in the 1970s and 1980s.

Answer 2

If you are asking about "push factors" that made Jews wish to leave Russia when that was not possible and then escape en masse in the early 1990s when they could, such things included: Workplace discrimination, illegality of their religion, small massacres, deplorable conditions in Russia, inability for social advancement, and being treated as unwanted visitors.