How did the United States get involved in Afghanistan gulf war?

There is no "Afghanistan Gulf War."

If you are referring to the Persian Gulf War, here's the detail. (Note that the Persian Gulf does not border on Afghanistan.)

Saddam Hussein, then leader of Iraq, sent troops to take over the neighboring small nation of Kuwait on August 2, 1990. His actions were denounced unanimously by the United Nations Security Council. On August 9, the United States sent forces to the Persian Gulf to begin a defense of Kuwait. By November, the UN Security Council resolved to use force against Iraq unless they withdrew from Kuwait by January 1, 1991. Since that did not happen, an international military effort was undertaken against Saddam Hussein's forces. The coalition forces, with troops coming from 32 nations, was under the supreme command of General Norman Schwarzkopf of the United States.

If you are referring to the US going to war in Afghanistan, that happened after terrorists attacked the United States on September 11, 2001 (known as 9/11). The terrorist group that launched the attacks had been operating in Afghanistan, and the United States went to war in Afghanistan initially to go after that terrorist group, al-Qaeda.