Why do people carry a palm tree branch on Palm Sunday?

Right Here there is an excellent explanation.

The palms are a representative reflection of the welcome that Jesus received from his followers as he entered into Jerusalem before being arrested and crucified.

It’s one of the many rituals in which something is done to symbolize something else. When I was a child those rituals seemed novel to me, but as I got older, frustration with the numerous, seemingly pointless rituals was one of the reasons why I turned away from Catholicism.

The explanation provided by Sloane2024 was good, not bad for Wikipedia!
I would like to add that as they lay those palm branches before the Messiah, it was a form of obescience as the righteous and, or saints are represented at times as trees in the scriptures.
For example: Isa 61:3 To appoint unto them that mourn in Zion, to give unto them beauty for ashes, the oil of joy for mourning, the garment of praise for the spirit of heaviness; that they might be called trees of righteousness, the planting of the LORD, that he might be glorified.

And: Psalm 1:3 And he shall be like a tree planted by the rivers of water, that bringeth forth his fruit in his season; his leaf also shall not wither; and whatsoever he doeth shall prosper.

They are just reinacting Jesus’ entrance into Jerusalem, when folks laid down cloaks and tree branches to cover his path, a mark of respect for someone who is worthy of adulation. The tree branches are assumed to have been palms, since Jerusalem is in the Middle East.

People used the palm leaves to lay on the ground when Jesus was riding the colt into the city. I guess the road was hot and the leaves took some of the heat of the colts hooves and made it an easier ride for Jesus,