In 11:1-10, Paul picks up on a common theme in the Hebrew Bible: there always a remnant of righteous within the unbelieving Israel. At the time of Elijah there was a remnant of faithful Jews who refused to worship Baal. When Isaiah is called to announce the coming exile he was told there will always be a “root in the stump of Jesse” which remains faithful. This remnant does not deserve to be preserved since they are as guilty of rejection as the rest of Israel, but they receive God’s grace nevertheless.
Paul says something like this on Cyprus, in Acts 13, when he blinds the Jewish sorcerer Elymas (blindness lasts for a short time). The belief that there is a righteous remnant within Israel must have been an encouragement for Paul to continue his preaching to the Jews even until Acts 28.
Israel’s stumbling is salvation for the Gentiles (Romans 11:11-24). Salvation came to the Gentiles in order to make Israel jealous and their sin makes possible riches for the Gentiles. The Gentiles therefore have no right to boast to the Jews because they are like branches grafted into a tree. If God did not spare the natural branches (Israel) he will certainly not spare the grafted-in branches (the Gentiles).
The falling away of Israel and the subsequent offer of salvation to the Gentiles demonstrates two attributes of God that might be thought of as contradictory, justice and mercy. By judging his people he has made room for the Gentiles, who by the mercy of God are allowed to participate in God’s grace through faith.
But Paul also indicates Israel will yet be saved in the future (11:25-32). Paul calls this future restoration of Israel a “mystery,” something not previously revealed. The specific content of the mystery is that Israel is experiencing hardening until the full number of Gentiles has come in. (11:25-27). How this salvation happens is a dividing point between premillenialists, who anticipate some kind of real restoration of Israel, and amillenialists, who would see the restoration only through the Church.
The reason for this restoration is that God’s promise to them is irrevocable (11:28-32). The Promise made to Abraham was unconditional, God was going to make a people for himself, and no amount of unfaithfulness on the part of the nation of Israel would prevent that plan from happening.
The main point of all of this for Paul is God’s glory. (11:33-36). Paul say God will receive all the praise and glory for restoring his people Israel, despite their rejection of the Covenant and the Messiah.