A big part of what Dawkins and others do is signalling high status of atheists.

To explain, imagine that you have two groups A and B living next to each other. Members of the group A have regular meetings listening to guys who say offensive stuff about the group B. ("They are fools, they are immoral, they will be punished in the future and they deserve it.") Speaking these offensive things is socially acceptable; actually it is kind of a taboo to point out their offensiveness. The freedom of group A to speak offensive things about group B is considered one of the most important rights in the society. -- On the other hand, members of the group B are taught that in privacy they are allowed to politely disagree with the teachings of group A, but in any confrontations being extremely polite is critical to their being good citizens. No matter how unfair the group A is towards them, the group B is not socially allowed to defend on the same level.

Guess which one of these two groups has a higher status.

Maybe the group A are Muslims, and the group B are dhimmis. Or maybe the group A are Christians, and the group B are atheists.

People like Dawkins are demonstrating that it is not only okay to be an atheist, but it is okay to be an atheist who does not behave like a dhimmi, despite being socially conditioned and expected to do so. They show that you can display high status and get away with it. They show that the balance of power is not the same as in the past when the social norms were formed. Your religious neighbors may be offended, but they can't burn you at the stake anymore. And fuck them; they were saying and listening to offensive stuff about people like you for years, you were just not socially expected to fight back!

The relation between this and conversion is indirect. In a perfect world, people would choose their religious opinions based on evidence. In this world, status plays an important role in choosing sides. No one has a desire to become low-status, but some people are willing to accept this trade-off because of other things they value. Without strong reasons to act otherwise, people are most likely to choose the higher-status group and remain loyal to it. Now more people can become atheists without paying a cost in status.

And yeah, being right means that now you have status and reality on your side. Which is still far from being an obvious winner, because magical thinking and other biases remain on the other side. Anyway, having more status is better than having less status, ceteris paribus.

Religious apologetics also contains some status-building parts (it often says that people are irreligious because they are stupid and evil), but for the religious side that's not the only channel for status signalling. There is nothing offensive about atheists that you wouldn't already hear repeatedly in most churches. So I guess it is safe to assume that religious apologetics' primary function is something else.