What does Luke 5:37 mean?

Jesus continues His explanation of how His way of salvation—the new covenant—cannot coexist with the rigid traditions that have crept into the Judaism of the time. He has already explained how their religious traditions can't be "fixed" by just including some of what He teaches (Luke 5:36). Here, Jesus says they cannot practice the deeper truths of His way and maintain the appearance of their way. Christianity cannot be contained within a rigid view of Jewish tradition.

A wineskin was made by taking the hide of a sheep or goat, removing the hair, and treating the hide. The holes in the body were sewn up, and the neck was used as the neck of the bladder. When new wine was put in the skin, it expanded as it fermented, stretching the new hide. If the hide was old, it became brittle and could no longer accommodate the expansion of new wine. The hide would crack, thus ruining it, and the wine would spill out, thus wasting it.

In this symbolism, the old skin is the religion practiced by the Pharisees, including added fasts (Luke 5:33). That cannot coexist with the new wine of the new covenant. In the context of Luke 5:33–39, the gospel precludes mournful fasting while the Messiah is present.