What does Romans 10:20 mean?

Israel should have understood that God would eventually welcome non-Jews into relationship with Him. That's the case Paul is making about their rejection of Jesus as the Messiah. In the previous verse, he pointed to God's words in Deuteronomy that he would make Israel jealous of those who are not a nation, just as they had symbolically made Him "jealous" with their worship of non-gods.

Now Paul quotes Isaiah, again by name. Isaiah's bold statement also quotes God's own words. Paul claims them out of the context of Isaiah 65:1 and applies them as an analogy of how God has rescued the Gentiles. He has been found by those who didn't look for Him. He has shown Himself to those who didn't ask.

This circles back to Paul's question at the end of the previous chapter: "What shall we say, then? That Gentiles who did not pursue righteousness have attained it, that is, a righteousness that is by faith; but that Israel who pursued a law that would lead to righteousness failed in reaching that law" (Romans 9:30–31).

God had declared believing Gentiles righteous based only on their faith in Christ. He did this even though they had never tried to become righteous through their own good deeds or by following the law. At the same time, God had rejected Israel's attempt to become righteous on their own without faith in Christ. (Romans 3:20).