What does John 12:13 mean?

Palm branches were ancient symbols of victory. The term hosanna was most famously used in Psalm 118:25–26, and means "save now!" It came to be a figure of speech praising God for deliverance, and it was said in support of a king or victor. So, the people who celebrate Jesus' entry into the city are deliberately applying Old Testament ideas to Him. They are proclaiming Him as the Promised One and the Savior of Israel. Jesus, for His part, will deliberately fulfill certain aspects of Old Testament by riding on the back of a donkey (John 12:14–15).

Daniel 9:25 also speaks of the coming Messiah. That passage implies that after a certain event, there would be a set time ending with the arrival of the Promised One, who would then be "cut off" (Daniel 9:26). Using the prophetic concept of days and years, this predicted 173,880 days. Starting with the decree of Artaxerxes to rebuild the walls of Jerusalem on March 5, 444 BC, Daniel's predicted timeline ends on March 30, AD 33 – the Monday before Passover (John 12:1; 12:12).

Unfortunately, many of the people still misunderstand the nature of Jesus' role. Those who turned away after He fed thousands (John 6:26) were disappointed that He spoke of spiritual issues, instead of political ones. It's likely that most of those cheering for Jesus in this crowd are also expecting Him to come as a conquering king, to oppose the Roman oppressors.