Romans 11:13-16 meaning

The Romans in Paul's audience are mostly Gentiles; he is speaking to you who are Gentiles (v 13), In particular, these are Gentile believers whose faith is spoken of throughout the whole world (Romans 1:8). He continues to defend against allegations slandering him as if he taught that there is not a future place for Israel. Paul strongly asserts that God's promises to Israel remain fully intact. 

Notwithstanding Paul's staunch faith that God will fully restore Israel, Paul desires the Israelites be reconciled with God now. He wishes Israel would take note of God's new favor on the Gentiles and call out to Him again (Romans 10:13). He says inasmuch then as I am an apostle of Gentiles, I magnify my ministry, if somehow I might move to jealousy my fellow countrymen and save some of them (vv 13-14). 

Even though Paul's primary ministry is to the Gentiles, his heart is still with his people. Paul continued to be fully Jewish throughout his ministry, telling the Jewish leaders in Rome, "Brethren, though I had done nothing against our people or the customs of our fathers, yet I was delivered as a prisoner from Jerusalem into the hands of the Romans" (Acts 28:17). Paul was ordained and directed by God to minister to the Gentiles as well as his people (Acts 9:15). After the Jews vehemently rejected his message, he began to focus on the Gentiles (Acts 13:46). 

Paul asserts that eventually the nation of Israel will be reconciled with God, just as their rejection of God has paved a way for the reconciliation of the world (v 15). And then, what will their acceptance be but life from the dead (v 15)? It will be like a resurrection from the dead; an amazing, miraculous, celebratory event. 

Therefore, Paul is asserting the exact opposite of the idea that God rejected Israel and rejected the Jews. Paul declares that God still loves Israel and intends to redeem them in every way. It ought to be the goal of Christians to bless the Jewish people and to encourage them back into their own fold. 

Once they are grafted back into their own natural tree, the result is even greater for all of us. Paul is definitively refuting the defamatory allegations made by the competing Jewish "authorities" that Paul teaches that God has rejected Israel. In fact, verse 16 makes it clear that Israel is the first piece of dough in the bread, and if the first piece of dough is holy, the lump is also (v 16). Israel is the root of the tree and if the root is holy, the branches are too (v 16).

Israel is the root of the olive tree into which Gentiles have been grafted, like wild olive branches, as the next passage will expound upon (Romans 11:17). Israel is also like the first piece of dough from which a larger batch of bread is made. All things Christian are rooted in that which is Jewish. Jesus is Jewish. He came to fulfill the entire Jewish Law (Matthew 5:17-18). All the promises to the Jews remain intact, and will be fulfilled. God's promises are irrevocable (Romans 11:29).