It's longer dive times. You can't dive deeper on Nitrox - that can cause O2 toxicity and kill you.
Because you're diving say, 36% O2 when air is regularly 21%, that means instead of inhaling 79% Nitrogen, you're only inhaling 64% nitrogen.
That means you can safely stay at 80 feet for much longer than you could simply breathing air, because you're not absorbing as much nitrogen, therefore causing less risk of decompression sickness.
Now, why don't we breath 100% O2 then and never have to worry about nitrogen? Because oxygen is toxic at depth as well. If you exceed any deeper than 20 feet or so on pure O2, your body will be absorbing too much Oxygen. You can start to spasm and have seizures and ultimately drown.
Now - there's a lot more to all of it than this, but that's it Nitrox in a REALLY simple nutshell. Of course, get proper training, et al.