There are two specific ways the Gospel has advanced as a result of his imprisonment. First, the whole “imperial guard” has heard Paul’s imprisonment is on account of Christ. The word Paul chose here is refers to the praetorium (πραιτώριον), or the headquarters of the Roman guard. In any major city with a Roman presence there might be such a headquarters, certainly Herod’s palace in Caesarea could be called a praetorium and “all the rest” are the people living in the palace. If Paul is in Rome, then the term could refer to the Imperial guard. In either case, everyone who has contact with Paul knows why he is in prison.
Second, “most of the brothers” have grown in confidence and are speaking the Gospel more boldly and without fear. Paul’s example of boldly preaching the gospel despite his chains has convinced timid believers to be more open in their faith. This verb (τολμάω) has the sense of daring to do something (as in Rom 5:7, someone might dare to die). Joseph of Armethea, for example, had the courage to approach Pilate and ask for the body of Jesus (Mark 15:43). This request was courageous since Joseph might have been seen as rebellious or revolutionary by Pilate, potentially he too could have been executed for providing an honorable burial for Jesus.
To be “fearless” has the connotation in contemporary English of doing something dangerous, such as skydiving or base-jumping. People put “fearless” on t-shirts (although usually not the people who jump off skyscrapers). But the word in Greek can have the connotation of shameless actions as well. To be brave and bold might mean that you do something dangerous and risky, but it also might mean you do something that is socially embarrassing (the old “truth or dare” game?)
It is possible some Roman Christians (again, assuming the traditional view) were in fact believers, but quite timid in making some sort of public proclamation of their faith. They were afraid to tell friends and family they were a part of this new religious group that worshiped a crucified man. Their belief in Jesus as the Messiah and savior would be shameful if they were Jewish or Roman and in either case could result in a loss of status, threatening their position in society.
In the Roman world, to boldly declare you are believer in Jesus was more than socially awkward, it was potentially life threatening! Paul’s imprisonment, therefore, is good because the Gospel continues to advance despite his chains. His example to other believers has emboldened them to publicly declare their faith in Jesus, even if that declaration is socially dangerous.
In many parts of the world, it is in fact dangerous to openly declare one’s faith in Jesus Christ. For people living in Southeast Asia, or example, to accept Christ is to reject the religion and traditions of your own culture. For people living in Africa, to accept Jesus is to reject the traditional religion of your family.
Does a contemporary American risk anything when they declare they are a Christian? I live in Grand Rapids, Michigan, the buckle of the Bible Belt. There is little risk in coming out as Christian in West Michigan compared to China, Vietnam, Iran, etc. Despite what you might think from watching news reports on loud American Christians, there is a great deal of timidity and fear among American Christians. Why is this? What do we have to lose by remaining timid in our faith?