Let's say you want to buy shares of a thinly traded stock. (I, a retail investor, experience this often when I buy preferred share issues, despite many being from big name companies with a very liquid market for their common shares.)
A level 1 quote would show you the lowest ask price, and indicate how many shares are offered at that price. If that's more than you want to buy, fine. Your market order will likely fill at that price, provided nobody beats you to those shares.
But let's say only 100 shares are offered at the asking price, and you want to buy 1000 shares. How could you know the likely average price you would pay if you entered a market order just then? (I don’t intend to imply that a market order is the best idea in cases like this.) Your order will most likely not fill all at the ask price you see quoted, because that seller (or a bunch of sellers with the same asking price) hasn't enough shares.
The answer to that question is: level 2 quotes. The example market order could, say, get the first 100 shares filled at the lowest ask, a few hundred more at the next higher ask (what price is that?), and the rest at some even higher ask (and what price is that?). If you want to know all of these likely fill prices in advance of your order being entered and filled, level 2 quotes give you the information you need — and not just the "tip of the iceberg" you would only see with level 1. Level 2 quotes can inform what limit price to use if you want a good chance at 100% fill but without the risk of paying more than you want. Limit orders (and marketable limit orders, if you aren’t that patient) are a better way to trade low liquidity issues.
I explain more about this in my answer to this other question: Can someone explain a stock's “bid” vs. “ask” price relative to “current” price?