How did the period after 1914 (up to the present) create an “era of American influence?”
This era was created most importantly by WWII and the Cold War. The time from WWI to WWII was important mainly because it led to WWII.
After WWI, the US pulled back into isolationism and was not very influential. But then US industrial might was one of the main causes of Allied victory in WWII. Because of this and because the US was the only country that was essentially untouched (in terms of its territory) by the war, the US became the most powerful country in the world. This combined with Western fear of communism to make the US the leader of the "free world" between 1945 and the end of the Cold War. This made that era an era of US influence.