Who was Jesse and how is he connected to Christ?

Jesse was the father of David, and Christ is considered the son of David. The connection runs from Jesse through David, Solomon, and the whole line of Jewish kings; Isaiah foretold a root of Jesse who would rise to reign over the Gentiles, in whom Gentiles would trust.

Les explains that Jesse was the father of David, and Christ is considered the son of David. So the connection goes all the way back from Jesse through David, Solomon, and that whole line of Jewish kings. He also points to the prophecy that there shall be a root of Jesse, and he that shall rise to reign over the Gentiles, and in him shall the Gentiles trust. Les notes this has not yet happened, but it is going to.

Isaiah 11:10

And in that day there shall be a root of Jesse, which shall stand for an ensign of the people; to it shall the Gentiles seek: and his rest shall be glorious.

Has the prophecy that the Gentiles would trust in the root of Jesse been fulfilled?

The prophecy concerning the Gentiles trusting in the root of Jesse has not been fulfilled. According to the context, it never happened in history, but it is going to happen. The root of Jesse refers to Christ, the son of David, who will rise to reign over the Gentiles and in whom they will trust.