IIRC, the US got involved at the behest of British Petroleum (BP).
I believe BP owned basically the entirety of the oil industry in Iran and ran it like the company towns of old. The living conditions were bad, the pay was worse, and the work was as dangerous as it was backbreaking. So when a beloved leader was democratically elected partially on the pledge of kicking BP out of Iran - which he did by nationalizing the oil industry - BP came crying to the US. Who were all too happy to destroy the biggest democracy in the Middle East in the name of oil.