It was the fulfilment of the prophecy in Zechariah 9
Zechariah 9:9 Rejoice greatly, O daughter of Zion; shout, O daughter of Jerusalem: behold, thy King cometh unto thee: he is just, and having salvation; lowly, and riding upon an ass, and upon a colt the foal of an ass.
Thus, began the final week of Jesus’ ministry which would end in his crucifixion.
It was a sign to those who knew the scriptures, that Jesus was indeed the king promised by God.
Matthew 21:5 Tell ye the daughter of Sion, Behold, thy King cometh unto thee, meek, and sitting upon an ass, and a colt the foal of an ass.
John 12:15 Fear not, daughter of Sion: behold, thy King cometh, sitting on an ass’s colt.
The Jewish leaders still refused to believe and ended their week saying: ‘We have no king but Caesar’.
I hope you find this helpful.