So too at the present time there is a remnant, chosen by grace.  But if it is by grace, it is no longer on the basis of works; otherwise grace would no longer be grace. (ESV)
God always has a remnant. In every generation there are a people set apart for God's glory. As Paul showed us in the previous verses Israel had a remnant of people who had not bowed their knee to false idols. God had to reassure the great prophet Elijah about this remnant. In verse 5 of our text Paul speaks of a remnant. He prefaces his words about the remnant by saying, "at the present time."
Paul is saying that there continues to be a remnant of those who belong to God, but clearly something has changed. What is the difference? Grace. In the past those who were followers of God were known by their works. They were seeking to obey God's Law. They lived by the Law. This new remnant of believers is set apart by grace.
Paul was preaching to people who had placed their faith in Jesus Christ. No longer would they look to the Law for salvation. They knew that their works would not be sufficient for their salvation. Only the finished work of Jesus Christ on the cross would be able to save them. This was true for Jews and Gentiles alike. If there were to be a Jewish remnant it would have to be on the basis of grace.
I love how Paul re-emphasizes that if it is by grace, then it is naturally no longer by works. This sounds extremely obvious, but we have to realize that the Jewish faith was so firmly attached to righteousness by works that this idea of grace was a shock to the system. They would constantly need to be reminded to place their faith in Jesus' work and not their own works.
Just like the Jews of Paul's day, we all are tempted to place merit in our own efforts. Ask the Lord to help you to look to him alone for your righteousness.