When the Almighty offered the Torah to the nations and only the Children of Israel accepted it, there must have been some members of the other nations ready to accept the Torah.
Converts may be the souls of those nations that were ready to accept the Torah.
But when you deal with it in a more concrete way, it is not mystical.
Basically it is a ceremony of adoption.
The convert is getting into the family, not only getting into the religion.
Nobody can get into the Jewish religion without getting into the family, so it’s an adoption.
Halachically as well, the proselyte, when he prays, speaks about the God of our fathers Abraham, Isaac, and Jacob.
–Rabbi Adin Steinsaltz