Why did the United States and many other nations step in to the First Persian Gulf War?
The reverse the invasion of Kuwait.
The Iraqi Government led by Saddam Hussein had invaded Kuwait because he owed Kuwait a great deal of money that he borrowed for the Iran-Iraq War and its aftermath. He misinterpreted a conversation with a U.S. diplomat to mean that the U.S. would not oppose an invasion of Kuwait.
He also had a very large, well armed Army. Rather than cut back on his military after the Iran-Iraq War and disband his troops to fuel unemployment and potential civil unrest, he kept the Army large. He needed to pay for it. He needed to maintain control of the Iraq nation (no easy task). Invading Kuwait solved these problems it a Real Politic kind of way. It made use of his Army and eliminated his debt.
The United Nations and the United States put together a coalition of states opposed to the Iraqi regime and took back Kuwait but did not knock Saddam out of power. They needed Saddam to keep Shiite Isamists, sponsored by Iran, from making a successful rebellion or Kurdish rebels from making a break away state. Both of which would create regional instability.