Big institutional investors buy and sell large amounts of stock in a variety of ways (through different brokers, using different procedures, tempos, etc.). They are interested in judging the ex-post "quality of execution" , i.e. how good a price they are getting when buying and selling. Even being consistently off by a few pennies could be very costly in the long run...
Because purchases and sales are made throughout the day (not just at the close), a comparison of the price you bought with the closing price printed the next day in the WSJ is not good enough. Instead, consultants and data vendors compute the VWAP for every stock and every date, and when an institutional investor come to them with the dates, tickers and prices of their trades, they compute the average deviation from VWAP (for buys and sells separately). These should theoretically be zero, buying at a higher price than VWAP shows you bought "worse than randomly" and can be considered a measure of the "cost of trading" (bid-ask spread) and "market impact" of your trades. With this ex-post information, you can refine your procedures, fire the brokers who did a poor job, etc.
A second use of VWAP is in Algorithmic Trading. Algorithms exist that break up single large trade into smaller trades. These dynamic algorithms often rely on the current VWAP (the VWAP for all of todays trades up to the present time) to affect the rate of buying or selling. They might accelerate the pace of buying if the current price is below VWAP for example, and vice versa. (These are just heuristics with no claim of optimality of course). At the end of the day they can also use the day's VWAP to judge how well they did.