The symbolism of this mitzvah are:
1. That the Jewish people will endure suffering at the hands of other nations – including our brother Eisav – but that we will always persevere and never disappear. This message is driven home by attaching it to a mitzvah.
2. Rav Samson Rephael Hirsch explains that although the spirit of Eisav will never conquer Yaakov and his descendants, Eisav will be able to hamstring Yaakov and prevent him from standing firmly on two feet. Thus Yaakov goes through history with an infirm physical stand and gait. By having to remove the gid hanasheh, whenever Yaakov’s descendants sit down to eat meat, they realize that their continued existence is not dependent on their physical strength and stamina, but on spiritual factors which can never be weakened by Eisav’s might.
3. The Zohar explains that the thigh is the place from which all sexual desire extends. And there the angel was able to wound him. In every other struggle with the evil inclination we can overcome our evil urges if we so desire except where the lust is so strong that even great people are rendered powerless. Its very name means “to forget,” because once it has been aroused, all rational thinking and religious scruples are left far behind. The only way to win that war is to stay far away in the first place, For this reason, the gid is not eaten at all but utterly avoided.
4. Rabbi Yehuda HaChassid (1150-1217) wrote in Sefer HaChassidim §231 that after Jacob was injured in his wrestling match with Esau's ministering angel, his sons felt guilty for having left their old father alone, so they accepted a promise on themselves not to eat the sciatic nerve of animals, to sympathize with their father's pain.