Why was Paul's body suffering while the people he preached to were being saved?

Paul's body suffered because he expended his physical life and energies to get the gospel to pagan people, who were coming out of spiritual darkness into the joy and life of the gospel. His bodily suffering was the consequence of that labor, and he endured pain in prison for the gospel's sake.

Les explains that Paul's bodily suffering was the direct result of his sacrificial labor. He spent his physical life and energies getting the gospel to the pagans, and through that ministry they were coming out of their pagan darkness and spiritual darkness into the joy and life of the gospel. While they were coming into that gospel life, his body was suffering the consequences. Les pictures him as an old fellow wracked with pain in a Roman prison, waiting for the executioner's sword, because he had been through so much for the sake of the gospel. The same spirit of faith appears in verse 13: it does not say he did anything else but that he believed. The Acts context shows the great change of direction from Peter and Israel to Saul of Tarsus, the persecutor of Jewish believers.

2 Corinthians 4:12

So then death worketh in us, but life in you.

Why was Paul's body suffering while the Gentiles he reached were coming into the gospel?

Paul's body suffered because he spent his physical life and energies bringing the gospel to pagan Gentiles, who came out of spiritual darkness into the joy and life of the gospel. He endured pain, imprisonment in Rome, and hardship for the sake of the gospel, suffering the consequences of his sacrificial labor.