King Anti-Nephi Lehi makes it clear in his sermon that the miraculous conversions did not come completely out of the blue. Indeed, ironically it was the wicked priests of King Noah that might have opened the way for the ultimate conversion of many Lamanities.
His people were Nephites who then went and joined the Lamanites. In doing so, the process of opening up a correspondence with the Nephites began. This involved the teaching of the Nephite language and culture. So it was the wicked priests who broke down barriers to allow the true gospel to be taught. The Lord works in mysterious and profound ways.