How do the presence of martyrs in a culture lead to conversions of the faith?
I think the martyrdom stories that got circulated were very important for the development of early Christianity. Several of the martyrdoms talk about, of course we don't know how to judge the historical veracity of those tales, but they do say that there were pagans present at these martyrdoms who were so impressed by the courage of the Christians that they came to see the truth of the Christian religion themselves and immediately converted to Christianity. Probably, for the most part, though, these martyrdom accounts were written for other Christians to try to bolster the Christians' faith at a time of persecution. To keep up your courage in case this happened to you as well.
Rather than saying that persecution leads to church growth, it may be more accurate to say that, as we look at the world today, we find that church growth almost always leads to persecution of some sort. The church is growing worldwide at an amazing rate, often most rapidly in the very countries where persecution is the most severe. Very often, as in the case in Vietnam, Laos, Ethiopia, China, and India, the persecution is a direct result of the Church’s expansion. Seeing the growth of Christianity, governments and/or leaders of traditional religious groups feel threatened and deliberately attempt to control or stop the growth of Christ’s Church. As a result, people see that Christianity is something people are willing to die for and it leads to conversions.