The ass , when compared to a horse, is a lowly creature. The horse is more powerful, majestic, and bold. However, the horse is also used in war and was a symbol of victory when the king returned from battle riding high on his stallion. The lowly ass was a symbol of peace. Therefore, when Jesus rode in on an ass, He was coming in peace, not in war as the leaders of the day feared.
Why didn't the people understand?
The leaders wanted to keep the status quo. They wanted to stay in power and were determined to do anything to remain there. The people wanted change. They were tired of the Roman government and wanted their Messiah to come swooping in and obtain a physical, here-and-now victory over their enemies. All these ideas and prejudices blinded them to the truth--Jesus was here for all of them. He was entering the gates of Jerusalem as the first step to giving them what they most needed and, in truth, desperately wanted. He was giving Himself.
Just a few short days later, these same people that cheered and greeted Him at the gate turned on Him with venom and demanded that He be put to death. Unbeknown to them, they didn't really succeed in killing Him. He freely and willingly gave His life for all those people there, for all the people before them and for all the people after them forever. There was no way they could take His life if He didn't offer it to them.
That's why He rode on an ass in peace. He was peacefully taking our place at our judgment and sentencing.
That's why the palm leaves were unknowingly chosen by the people. They were forecasting a triumph that, at the time, even they couldn't possibly understand without truly believing in their hearts what He has said.
He triumphed over sin, death, and hell not by taking lives in a war, but by giving His own in our place.