Every year my mother would try to remember how to weave her palm from Palm Sunday. It was something she had learned when she was young. I remember the braided palms hanging behind pictures in the house and feeling a sense of comfort knowing that they were blessed palms. They weren’t decorations, they are a sacramental. A sacramental is something that is blessed and lead us to the sacraments of the Church. Also, the palms remind us of the suffering of Christ.
Palm Sunday is the day that Jesus came into Jerusalem and was treated as a king and prophet but a week later he would be crucified. This is the beginning of his Passion or suffering that he endured for us.
Since they are blessed, you can’t throw them away. To dispose properly of the blessed palms, either burn them and dispose of the ashes in your lawn or garden or bury them in a respectful manner. Also, you can bring them to the Catholic Church and they will burn them to make ashes for Ash Wednesday.
This link has some great resources on Palm Weaving. The link on the bottom for the “Lost Art of Palm Weaving” is an excellent resource. The nun did a lot of research on palm weaving and created a book on the subject before the Art was lost forever. https://www.catholicculture.org/culture/liturgicalyear/activities/view.cfm?id=1035
Video credits: Arlington Catholic Herald