What does it mean that Paul was being brought back under bondage?
Being brought back under bondage means returning to the Law—circumcision, dietary regulations, and the Laws of Moses. The Judaizers wanted Paul to concede that his gospel preaching was wrong and to instruct his congregations to adopt these legal requirements, effectively reversing the liberty he preached among the Gentiles.
Les explains that the Judaizers, though mostly believers in Christ as the Messiah, came in secretly to spy out the liberty believers have in Christ Jesus. They watched Paul and Barnabas closely, especially regarding Titus, to see if they practiced circumcision and kept kosher. Their goal was to submarine and sink Paul, to put him out of business rather than let him go to the Gentiles with his message. They wanted to bring Paul back under bondage, which means the Law. They wanted Paul to say, "Okay fellows, you win," go back to his congregations, admit he had it all wrong, and tell people to go ahead and practice circumcision, keep the dietary Laws, and keep the Laws of Moses.
And that because of false brethren unawares brought in, who came in privily to spy out our liberty which we have in Christ Jesus, that they might bring us into bondage: