This passage is not just about the Jews. It’s about me. It’s about you. All of us, before turning to Christ, have sought to establish our own righteousness. Like a child, we fight and say “Let me do it! I can do it!”. In our foolishness we pridefully try to work out our own righteousness so that we can stand before God under our own strength and goodness and say “I did it”.
But if you have lived long at all, the impossibility of it all comes crashing down around you. And this happened to the Jews as well. Trying to achieve a righteousness by following the law, they rejected the gift of righteousness (and thus rejected Christ). The irony is that, in trying to stand righteous and justified before God under our own strength and merit, we reject the very one God sent. We reject God himself. Through obeying the law no one will attain righteousness, but through Christ, who is the end of the law, there is “righteousness for everyone who believes”.
Are you still trying to "earn" God's favor when you do good things? Are you still trying to prove yourself to God?
Or are you doing good out of a response of gratefulness for what Christ has already done for you? Are you pursuing righteous living out of a love for Jesus and a desire to be more like him? Test yourself. Test your heart.