The basic idea is the same as regular composting, except red wiggler worms eat your organic waste and their poop -- sorry, castings -- makes some sweet, sweet fertilizer. Additionally, the water that drains out through your worm bin (or, via some other method, has had the castings soaked in it) is known as worm tea and is supposed to be great for watering your houseplants.
The worms are faster than regular composting, too. Depending on the ratio of worms to food scraps, usable compost is expected in 3 or 4 months. The big advantage is that it's workable for apartments... it doesn't smell if you do it right (i.e. bury the scraps under the bedding, don't add more than your worms can eat, etc.), it's covered, so flies are at a minimum (we'll see when summer comes around), and it fits in the corner.
The bedding of shredded newspaper and brown paper bags is ready, and the first batch of food scraps is added.
My worm wrangler and I adding some worms and castings my friend Marilyn gave me.