The event of Jesus triumphal entry is important because it reveals Jesus kingship and fulfills prophecy. Jesus was worshiped during the triumphal entry because people were proclaiming the sign He did in raising Lazarus from the dead and it seems because of the reputation of His ministry.
It also reveals Jesus is the true King and is a lion and a lamb, he conquered by dying to self. We also will be conquerors in the same way, by the same method. We will conquer sin and death through dying to ourselves and being made new by the power of the Spirit.