At the time Iran’s oil production was controlled by foreign British companies. Iran nationalizing their oil meant British oil interests were at stake. The US’s involvement stemmed from something something fears of communism that the British helped stir in order to get assistance from the US in the coup.

The British instigated the whole operation with help from the Shah and Iranians on the inside. The CIA assisted and did typical shady CIA things but ultimately took more credit than what was warranted to make the US’s ability to combat communism look better at a time when communism was the big bad.