For I am not ashamed of the gospel, for it is the power of God for salvation to everyone who believes, to the Jew first and also to the Greek. Romans 1:16
Imagine an vital outbreak had spread across the world and infected the entire population. People were losing their minds and one by one they were all dying off. All sorts of superstitions and false remedies were being peddled; blood letting, voodoo dolls, snake oil, relentless exercise and more. Yet none of these really worked. There is one absurd remedy a "supposed" vaccine that is so ludicrous and such an obvious scam that the people who peddle it are being killed.
In this environment you are yanked from your vehicle, pinned to the road and injected with this scam vaccine. For days you are blinded and confused only to eventually wake up cured from the disease. The vaccine worked! You take your stories to others and see that everyone who takes it is truly cured. As the word gets out that you are injecting people with this vaccine you are arrested. You find yourself before the courts facing prison and even death. Your resolve is unshaken, you find no shame proclaiming the healing power of the vaccine because you are cured and all who have taken it are cured as well. In the face of persecution your boldness grows. If you are silenced then millions will die. You are not ashamed of this vaccine because it has the power to bring life to those that are dying and to cure their disease.
This is where Paul was as he proclaimed that he was not ashamed of the gospel. He had tried to cure his sinful living with the relentless exercise of the law but found that it only increased the internal symptoms of his sin. Blinded by Jesus and given a revelation of the gospel he was cured. A new man with a new life. He spread the message and saw many others freed from their sin and given new life. As persecution, trials and imprisonment tried to silence him he grew bolder knowing that his message must get out. The cure was real and contained the power to save millions from certain death.