In this passage, Paul explains that a remnant of Israel is a believing remnant, while the majority of Israel has been hardened so as not to trust in the gospel.
He begins by saying that God has not rejected the Jews. The first proof for this is that the Apostle Paul was saved, and he was a Jew. However, in verse 2, Paul specifies who God has not rejected: those whom He foreknew. This implies that He did not foreknow Jews who are destined for destruction.
Paul then proves from Scripture that there has always been a believing remnant of Jews. He uses the example of Elijah, in which Elijah claims that he is the only believing Jew alive. However, God tells him that there are others who are believers, and thus, have not bowed down to Baal, an idol. In verse 6, Paul concludes that salvation is based on grace, not on works.
In the next verse, Paul also concludes that Israel did not obtain righteousness, but the chosen Jews did, while the rest had their hearts hardened so as not to trust in the gospel. Then, Paul proves from Scripture that this is the case. In verse 8, he quotes Scripture that says God gave the majority of Jews over to stupor, blindness, and deafness. In the remaining verses, he quotes Scripture as wishing unbelief upon the majority of Israel.