What does Matthew 12:12 mean?

The Pharisees are hoping to trap Jesus: to catch Him healing on the Sabbath and thus breaking their interpretation of the fourth commandment (Exodus 20:8–10). The problem is their trap supposes their interpretation of "work" carries the same level of authority as the law of Moses. Jesus is showing it does not (Matthew 12:9–11).

Presented with a man who has a withered hand, Jesus has asked a question: Who would not lift one of their sheep from a pit if it fell in on the Sabbath? In other words, who would allow their sheep to suffer instead of doing the work of pulling it out of the pit, even on the Sabbath? Jesus' question assumes that most everyone would choose to show mercy to the sheep instead of woodenly following the letter of the law, rather than the intent of it.

Now Jesus brings that point home: How much more valuable is a man than a sheep? That's why, He concludes, it is lawful to do good on the Sabbath. The good in this case is showing mercy to a person if it is in your power to do so. Jesus will demonstrate in the following verses that this is within His power.

Jesus puts the question a little differently in Mark's version of the story: "Which is lawful on the Sabbath: to do good or to do evil?" That question implies that choosing not to do good, in the sense of showing mercy on the Sabbath, might be considered evil (James 4:17).

It's important to understand that Jesus is not teaching that observing the Sabbath is wrong. Nor is He suggesting that the literal meaning of any law is unimportant. The Sabbath was one of God's great gifts to Israel. However, the requirement not to work was intended to bring God's people rest—not to add to their burden. Jesus is objecting to how the Pharisees have twisted God's commands. Again, He shows that they don't understand that God desires mercy, not sacrifice (Matthew 12:7).