The reference to dogs is to the snarling, half-wild curs found on the streets of the city at that time. Because of what these dogs fed on—garbage, decayed meat, rotten vegetables—they were regarded as unclean.
Paul is referring to Judaizers, legalistic Christians of Jewish descent, who taught that it was necessary to observe the law of Moses—especially circumcision—to be a real Christian. Sadly, these people are still with us. Paul says, like dogs they were feeding on the garbage of carnal ordinances. They were holding up, as of great value, rituals that once were valuable, but now are decayed, over-ripe and fit only to be thrown out. They were tirelessly seeking to convert young believers and bring them back under the bondage of legalistic restrictions.
This kind of teaching has a strong appeal to our human thinking because of its apparent show of devotion—some ornate, solemn ritual done in a religious manner—was worthy of some merit before God. It is so gratifying to the religious ego to perform some solemn ritual, and to be constantly busy at religious work, or even to mock the flesh in some way perhaps with a distinctive garb or an identifying posture. All of this, Paul says, is the enemy of true spirituality. It destroys the spirit of rejoicing, and it makes religion an empty, barren mockery. It puts the emphasis on the external, and removes it from the vital, the interior genuine aspects of faith.
The truth is we are saved by grace—and grace alone. Grace, God’s unearned, unmerited favor, is counter-intuitive to the world’s systems. But it teaches that we are to serve God and others out of gratitude and worship, rejoicing that He does His work in and through us. It reminds us that we can’t earn or merit the salvation that we enjoy, so we should quit trying.