The Catholic Encyclopedia says the following:
Palm branches have been used by all nations as an emblem of joy and victory over enemies; in Christianity as a sign of victory over the flesh and the world according to Psalm 92:13, "Justus ut palma florebit "; hence especially associated with the memory of the martyrs. The palms blessed on Palm Sunday were used in the procession of the day, then taken home by the faithful and used as a sacramental. They were preserved in prominent places in the house, in the barns, and in the fields, and thrown into the fire during storms. On the Lower Rhine the custom exists of decorating the grave with blessed palms. From the blessed palms the ashes are procured for Ash Wednesday.
It's a common custom for people to braid their palms or shape them into crosses, and then keep them in their homes through the year. (Being blessed, they shouldn't be just thrown in the trash.) These palms can help us recall Christ's victory year round.
The just shall flourish like the palm tree, shall grow like a cedar of Lebanon. (Ps 92:13)
On the next day, when the great crowd that had come to the feast heard that Jesus was coming to Jerusalem, they took palm branches and went out to meet him, and cried out: "Hosanna! Blessed is he who comes in the name of the Lord, (even) the king of Israel." (Jn 12:12-3)