It just doesn't
This is simply a lie, and I actually took a day off to trace it back to the person who started it -- it turns out they were trying to sell their own brand of deodorant
by claiming the others cause cancer
Frankly, you need to be able to recognize quacks. Do you think the folks at the NIH would use deodorant if it were unsafe? They're the ones who'd know. For your own survival, be discerning.