The word "Prince" is translated from the Greek archegos, which is translated "author" or "captain" in Hebrews 2:10. But here, Jesus Christ is called "the archegos of life, whom God raised from the dead, of which we are witnesses." In this context, the word has the sense of being "originator," someone who starts or begins something. An archegos is one who leads the way so that others may follow. It can also be translated "trailblazer," "scout," or "pioneer," and so it indicates one who leads into battle, blazes a trail, sets a pattern, one who initiates and guides.
In the Daily Study Bible series commentary by William Barclay, he uses the illustration of a ship foundering on a rock. Someone jumps overboard with a rope and swims ashore, securing the line somewhere on the shore so that others are able to grab onto the rope and come to safety. The one who did it originally is an archegos. He fulfilled the role of an archegos.
That is what Christ is. He is saving us from the jagged rock, from the loss of our hope of eternal life. That is His job. He is leading and guiding us to the safety of actually being in the Kingdom of God.