Here's the five-cent explanation ...
Nitrox reduces nitrogen intake. It does this by replacing nitrogen in the gas you breathe with oxygen. It isn't that you need more oxygen, or that adding oxygen benefits you in any way ... it's simply that oxygen is the second most abundant gas on the planet, and therefore makes an inexpensive alternative.
By reducing nitrogen you get to either (a) stay down longer, (b) reduce your surface interval, or (c) reduce the potential risk of a nitrogen-related malady.
Pick one ... or possibly two.
Mostly the benefit of nitrox will be recognized by someone who is relatively good on air consumption (and therefore is limited by no-decompression limits rather than air consumption) ... and/or someone who is planning to do multiple dives per day. It's especially useful on those diving vacations where you're planning 3 or more dives per day over multiple days.
If you're a heavy breather (time-limited due to air usage) or only planning one or two dives per day, it won't really be worth the expense.
In some parts of the world ... for example where I live in Puget Sound ... regular nitrox use makes sense for the experienced diver who wants to dive relatively deep and stay a while ... which implies someone who's diving either a large single cylinder (I like 119's and 130's) or doubles (I often dive dual 100's) ... in that it will allow me to do dives in the 75-90 foot range and stay on the order of 60+ minutes without running into deco. But for most recreational divers using standard AL80's, the benefits of nitrox is minimal except on those multiple-dive days.
Oh ... and nitrox ain't a deep gas. One of the trade-offs is that it increases your oxygen exposure, which is potentially harmful. One of the reasons for the training is to teach you how to manage that risk.
... Bob (Grateful Diver)