Peat moss is the common name given to dried sphagnum moss, which grows typically on bogs and rotten organic material. It is used as a soil conditioner because of its ability to hold enormous amounts of water in its cells- both living or dead- up to 16-26 times as much water to moss by weight! It is sold in compressed bricks. You have to fluff it out to a loose, dusty composition before mixing it, since the loosened moss is what is being considered in the equation. This is an unsustainable element of the mix, but it's only added when you first make your mix. You won't ever need to add peat moss again.
Vermiculite is a hydrous, silicate mineral that expands like popcorn when heated. The resulting product is a coarse, pebble-like stone, with cracks all throughout the pieces. Thanks to it's new, expanded nature, it can hold and store an incredible amount of water. It is also very light and fluffy. You know those little white things you see in potting soil that you always though was styrofoam? That's vermiculite.
Compost is the only part of the mix that includes any nutrients at all. The other two are mostly just water retention. Compost is made from completely decomposed organic material. This organic material is broken down to the point that the nutrients are now directly accessible by the plant roots. There are many kinds of compost: animal manure, decomposed plants, mushroom, and worm castings. SFG recommends that when you buy your compost in bags, you buy as many various brands and kinds as possible. If you buy all of one kind, it can hurt your plants by accidentally being "too much" of some certain element in the compost. If you buy all the same brand, odds are good that it all came from the same source (like the same dairy farm), and won't be as varied as you'd like.
The use of "Mel's Mix" lets you know and choose exactly what your growing medium is. The soil is arguably THE most important part of your garden. The soil you use becomes the food you eat, so it makes sense to spend the time and money in the most varied, nutrient-laden soil possible. It also prevents you from having to be concerned with such annoyances as soil pH, fertilizers, etc.