In verse 14, Paul is quoting himself in what he said to Peter in a rebuke, "If you, a Jew, live like a Gentile, where do you get off asking Gentiles to be more Jewish than you are?"
So, in verse 15, when Paul says "We ourselves are Jews by birth," I'm suggesting that he's still quoting what he had said directly to Peter.
"You and I, Peter, believe in Christ Jesus. Although we, as Jews, had been entrusted with the law, the understanding of how God wanted us to live, we now know that God has a way of including not only Jews but others, as well."
BTW, Carl R. Holladay, in Preaching through the Christian Year C, reminds us that Jews already knew that no one is justified by works of the law (e.g., Psalm 143:2; Habakkuk 2:4; Genesis 15:6).
Although these words may have been addressed to Peter, they are of course part of his argument he is using to counteract the attempts of the Judaizers who had followed him to the Galatian congregation and tried to convert the new Christians to Judaism. Paul is asserting that Christians do not have to become Jews in order to be Christians.
"It is Christ who lives in me...I live by faith in the Son of God....I do not nullify the grace of God; for if justification comes through the law, then Christ died for nothing."
Tangent from Holladay: We usually read "faith in Christ" in verse 16 to mean that we place our faith and trust in him. Some recent commentators have pointed out that this phrase in Greek is more literally translated to mean the faith that Christ has. Holladay sums it up, "This places greater stress on the work of Christ in our behalf than on our faith in our own behalf."