What does Luke 13:14 mean?
It is the Sabbath and Jesus has been invited to speak at the synagogue. However, the ruler of the synagogue becomes indignant when Jesus interrupts teaching to heal a woman with a severe spine deformity (Luke 13:10–13). This ailment is somehow the work of a demon (Luke 13:16). It is against the Mosaic law to work on the Sabbath, and the ruler seems to think healing is a type of work (Deuteronomy 5:12–15; Exodus 20:8–11).The ruler tells the people that they break the Sabbath if they come to be healed. No mention is made of the synagogue ruler criticizing Jesus. It's more likely the man is too intimidated by Jesus to call Him out personally than that Luke just doesn't mention it. But Jesus takes the attack against the people personally and speaks as if the accusation is against Himself as the worker. He compares His work of freeing the woman from her bond to someone who tends to animals (Luke 13:15–16).
The laws about the Sabbath are confusing. The Mosaic law simply forbids "work," and the only examples given are collecting firewood and selling things (Numbers 15:32–36; Nehemiah 10:31). The scribes spent many years refining the Mosaic law, adding to it and specifying what the vaguer laws meant. They call these rules the "Oral Law" and claim they are lesser instructions from God that Moses didn't write down. The laws defining Sabbath work are based on the work required to build the tabernacle. For instance, it is unlawful to trap an animal on the Sabbath because the Israelites had to procure hides for the tabernacle. But rabbis and scribes debated further: it is okay to trap a dangerous animal, like one that has rabies, because if the animal kills someone, the person can't observe the Sabbath.
When Jesus touches the woman and she straightens up, the synagogue ruler may interpret the act as the work of construction or completion, which is against the Oral Law. Jesus is far more pragmatic. Anyone in that synagogue would lead their thirsty ox or donkey to water. There is every reason He should be allowed to heal a Jewish woman on God's designated day of rest (Luke 13:15–16).