It starts with the fact that we all have desires. We may disagree on where they come from or why, but it seems safe to simply state that we have them as fact.
The argument assumes, too, that those desires fall into two categories. On one hand, we have innate desires. These are shared by all, regardless of circumstances. We all feel hungry or thirsty or tired, so we could accept that there are innate (or natural) desires for food, drink, and rest. On the other hand, there are conditioned desires -- things you only want if you've been taught to want the or are in the proper circumstances. Most people who have ever lived have not desired an iPhone or a sports car. Most have never even heard of them.
If you consider your conditioned desires, you'll find that some reflect real things you could achieve, and others don't. I may desire a fast car, and I could get one; but I may also desire to adventure in a fantasy world that I've read about, or use magic like Harry Potter, or live in medieval England as a valorous knight. Some conditioned desires are impossible. (Anyone that watches TV or has an email account has encountered conditioned desires. No matter how much I'd like to, and no matter what that email says, I don't have a recently-deceased Nigerian prince in the family.)
However, if you consider your natural desires, you'll find that they all correspond to something real. You get hungry, and there is food. You may not be able to get some right now, but it exists. This is the key to the argument - that, with introspection, you discover that all your natural desires (those not instilled by television, friends, books, etc.) have a real fulfillment.
If that's true, then if you have a natural desire for something beyond this world, there must be a fulfillment for it beyond this world.
Here is a more formal version, as composed by Dr. Peter Kreeft:
1) Every natural, innate desire in us corresponds to some real object that can satisfy that desire.
2) But there exists in us a desire which nothing in time, nothing on earth, no creature can satisfy
3) Therefore there must exist something more than time, earth and creatures, which can satisfy this desire.
Want more? Listen to Dr. Peter Kreeft explain this argument at http://www.peterkreeft.com/audio/23_desire/peter-kreeft_desire_.mp3. Dr. Kreeft also writes an explanation, and answers a few potential objections, in Argument from Desire by Peter Kreeft.
12-5-2014 update: Fr. Robert Barron posted his commentary on God's Existence and the Argument from Desire