Part of it has to be the simple fact that they want to be multi-ethnic. I mean, lots of churches like the idea of racial diversity but don’t actually pursue the goal. The key, I think, is actually trying.
And it may be that a large church is more likely to already have minority members, meaning a minority family hoping not to be the only minority family in the church will be more likely to visit a large church.
And a megachurch is more likely to accommodate a broad range of political beliefs, which helps the church be more attractive to minorities. Indeed, my experience is that as churches grow, they become more diverse in many ways — politically, economically, racially, etc. And their increasing diversity helps them grow more.
Of course, for long-time members, the diversity can be disconcerting. They remember the congregation having a certain comfortable uniformity. As they see the uniformity disappearing, they feel less at home. Large churches find ways to cope with these feelings while continuing to pursue diversity within Christianity.