The story
The slang sense referring to sexual intercourse was inspired by "Push It", a Salt-n-Pepa song of the 1980s. The song is ostensibly only about dancing but clearly also implies sex.
Etymology from Wiktionary, CC BY-SA 4.0.
push it
Today it means: To make an extraordinary or risky effort; to behave in a way which tests the limits; to take chances; to expect too much. Full definition →