Jesus has been traveling toward Jerusalem since Luke 9:51 (the conclusion of the Lucan narrative of the Transfiguration). The final approach to the holy city is marked with a third passion prediction (18:31-34), two scenes in the nearby city of Jericho (18:35-43; 19:1-10) and the parable of the talents (19:11-27) – the latter of which highlights a kingly figure coming to claim what was rightfully his. With this preceding, Jesus makes his entrance procession into Jerusalem.
As Culpepper  notes, “Entrance processions were a familiar ceremony in the first century. Numerous kings and conquering generals had entered Jerusalem over the years. Although the welcoming ceremony of a conqueror and the celebration of the return of a victorious general can be distinguished from each other, they share similar features. Paul Brooks Duff has summarized the characteristic pattern of an entrance procession as follows:”
In such Greco-Roman entrance processions we have seen the following elements: (1) the conqueror/ruler is escorted into the city by the citizenry or the army of the conqueror. (2) The procession is accompanied by hymns and/or acclamations. (3) The Roman triumph has shown us that various elements in the procession … symbolically depict the authority of the ruler. (4) The entrance is followed by a ritual of appropriation, such as sacrifice, which takes place in the temple, whereby the ruler symbolically appropriates the city.
As examples of this pattern, scholars cite Josephus’s account of Alexander the Great’s entrance into Jerusalem and Plutarch’s description of Antony’s entry into Ephesus:
Then all the Jews together greeted Alexander with one voice and surrounded him … [then] he gave his hand to the high priest and, with the Jews running beside him, entered the city. Then he went up to the temple where he sacrificed to God under the direction of the high priest
When Antony made his entrance into Ephesus, women arrayed like Baccanals, and men and boys like satyrs and Pans, led the way before him, and the city was full of ivy and thyrsus-wands and harps and pipes and flutes, the people hailing him as Dionysius Giver of Joy and Beneficent.
The entrance in to Jerusalem marks the end of the travel dialogue, marking a new section in Luke’s narrative.
You can read a complete commentary on this gospel here.