Antiperspirants work by stopping sweat from getting to the skin's surface by blocking sweat glands. The bacteria that live on your skin don't have any sweat to break down, so they don't make any smelly waste. Natural deodorant doesn't make you sweat less, but it does stop smells by using ingredients that kill the bacteria that cause smells or by covering them up with a scent.
Many natural deodorants use baking soda to help get the skin's pH up to 8 or 9, which kills bacteria.