At Not Exactly Rocket Science:

The latest and most intriguing clue comes from Enrica Audero from the European Molecular Biology Laboratory, working together with European researchers and a special strain of mice. Audero has shown that altering the balance of the signalling molecule serotonin in the brainstems of mice can lead to sudden demise, in a way that resembles the unexpected death of SIDS babies. The mice spontaneously go through "crises" where their basic body functions like temperature control and their heartbeat go haywire.

Audero’s work builds on research published by an American group two years ago, which first suggested that SIDS is the result of faults in the way our brains reacts to serotonin. This signalling chemical helps to control the core functions of our body that lie outside the realm of conscious thought. It’s the serotonin system that lords over our heartbeats, breathing, sweating and shivering, while our brains are busy processing blogs or solving Su Dokus.

Through a series of post-mortem exams, David Paterson showed that SIDS babies have more serotonin-releasing neurons, but a lower density of serotonin receptors – protein docks that the molecule sticks to. It was a start, and Audero capitalised on it by showing for the first time how an altered serotonin network could actually lead to sudden death. 

The results are a bit of a surprise. After all, mice can survive quite happily if they are born without any serotonin neurons at all, even though their behaviour may be different. Audero isn’t suggesting that human SIDS babies have extra serotonin receptors; you’ll remember from Paterson’s study that they actually have a lower density of these. But the results do suggest that an imbalance in serotonin signalling can be life-threatening. Even without any overt outside shock, it’s enough to trigger the sort of sudden failure in core body functions that is typical of SIDS. The means may be different in humans, but the ends are the same.

Read the link for more. That may explain how "Back to Sleep" works: on their backs, babies are less likely to encounter suffocation hazards, and thus the "crises" passes without causing any real harm, instead of occuring while the infant is in a dangerous position.