What a crappy setting.
If both their and our world have magical energy, then just as their world, our world would have some gods as well.
If they intended to leech off some magical energy from our world due to shortage or whatever, it would be a diplomatic problem, since essentially, without the consent from the overseeing gods of our world, their actions would be listed as plundering strategic resources.
If the author wanted to give the transmigration a logical reason, at least he should think it through properly and not shrug it off with a few random sentences.