What does Romans 11:10 mean?

Paul is quoting Jewish Scriptures from the Old Testament to support the idea that God has often hardened some people, in terms of their ability to understand and believe Him. Romans chapter 9 took an in-depth look at how, when, and for what purposes this can happen. This verse continues a quotation from David's Psalm 69, often taken as being about Christ Himself. In that way, this quote from Psalm 69:22–23 can refer to the enemies of Christ.

David prayed that his enemies and, by Paul's implication, the Jewish unbelievers of his day, would have darkened eyes so they could not see or understand. That is, they will be hardened to the truth that Christ is the Messiah and, thus, refuse to have faith in Christ and be saved by God's grace.

It's not clear what is meant by their backs being bent forever. Those who reject Christ are, ultimately, facing an eternity of separation from God because of their stubbornness (John 5:40; Isaiah 6:9–10). Paul might be referring to the danger faced by all who refuse to come to Christ, which is to be subject to shame and suffering "forever."