Why do this? Apparently, it results in a richer and more-intense malt flavor, with less harshness compared to other methods, according to Strong; he also mentions that the resulting beer is also a bit deeper in color, with lower acidity. This may not be a technique you'd use for a hop-forward beer, such as an American IPA, but for malt-forward beers, such as Scotch Ales, it sounds like something worth trying.
One thing to keep in mind here is that with this method your efficiency will obviously take a big drop, which means that no-sparge brewing is more expensive. Since you're adding water to your first runnings to get up to your boil volume, as compared to lower-gravity second runnings from a sparge, you would have to use more grain than normal for your mash. Strong suggests measuring the gravity and volume of your first runnings, which you can then use to calculate your efficiency for future no-sparge brews. This basically means your first attempt will be a bit of a guinea pig-approach. As for this first attempt, Strong writes that a 1/3-increase in your grist is a good place to start.
A slightly different no-sparge method is suggested by John Palmer, another very well-respected and well-read homebrewer. Palmer's approach involves adding the water that you would normally use for the sparge (roughly 1.5 times the water used in the mash) into the mashtun at the end of the mash. The full amount in the tun is then drained into the kettle, and should equal your target boil volume. This method has the benefit of requiring less grain then Strong's approach (approximately 25% more than the standard grain bill, instead of Strong's 33%). However, the disadvantage is that your mashtun would obviously have to be large enough to hold the entire amount of brewing liquor and grain; maybe not a concern for small (lower-ABV) beers, but could easily cause trouble with bigger, or even moderate-strength beers for those of us with 10-gallon mash tuns. You would also have to calculate the temperature of the additional water to add, so that you don't exceed 170 F, which may lead to leeching of tannins into the wort. For Palmer's full article (and some handy calculations you can complete to make things easier), check it out here at the Brew Your Own site.