Romans 11:2-6 meaning

Paul references the time of the prophet Elijah, in the Old Testament, where Israel was overrun with paganism, murder, and hatred toward God: Or do you not know what the Scripture says in the passage about Elijah, how he pleads with God against Israel? "Lord, they have killed Your prophets, they have torn down Your altars, and I alone am left, and they are seeking my life" (vv 2-3)." 

Paul references here an episode in the Old Testament where the prophet Elijah was demoralized, and prayed to God in despair. Elijah believes all hope is lost, and he is as good as dead (1 Kings 19:10). God spoke to Elijah to let him know that he was not alone, that there were seven thousand men who were still faithful to worship Him. 

God did not throw away His chosen people. God has not rejected His people whom He foreknew (v 2). Paul continues his recall of scripture: But what is the divine response to him (Elijah)? God told the despondent Elijah, "I have kept for Myself seven thousand men who have not bowed the knee to Baal (v 4)." God reserved seven thousand men who stayed faithful to Him, as a remnant that God Himself set aside. The point is that there has been and always will be a remnant who will be faithful to God. 

Paul is saying, even now, God is not finished with Israel; He always keeps and nurtures a remnant that will follow Him faithfully: In the same way then, there has also come to be at the present time a remnant according to God's gracious choice (v 5). This is all done through God's grace, not because someone "followed the rules" resulting in God somehow becoming obligated to perform for them. 

But if it is by grace, it is no longer on the basis of works, otherwise grace is no longer grace (v 6). On the contrary, no one deserves His favor, or can ever demand favor from God. God is God, God is not a cosmic vending machine we can manipulate, in any way. Israel has rejected God, but God has not abandoned Israel, and He will not let Israel slip away from Him entirely.

Grace means favor. When God gives favor, it is never because someone has lived up to a standard external to God. That doesn't mean that His grace is not given without reason. God does what He does because He chooses to do it. He favors people because He chooses to do so. We never deserve His mercy, in the sense that we can say 'We did such and such therefore…" but He gives mercy to whomever He chooses (Romans 9:15). 

It is interesting that Paul says God "foreknew" His people, referring to Israel, using the same word he used to describe all believers (also His people) in Romans 8:29. When God promises something, He will always do it (Romans 11:29). God promises in Romans 8:29 that believers in Jesus will be conformed to the image of His Son. God will cause all things to work together for good, even difficult circumstances. That even applies to mistakes we make. 

God will redeem the bad choices Israel makes, just as God causes "all things" to conform believers to His image. The question is not whether, but how and when, and how many negative consequences we choose for ourselves. Israel is incurring a lot of pain that could have been avoided. Israel might be straying right now, but God still has a plan to restore Israel. In fact, similar to the promise of Romans 8:29, Paul will assure us in Romans 11:26 that all Israel will be saved.