The key benefit is that the fuel isn't burned. It's converted chemically, so it is a very efficient process.
They have a good flash video explaining it here: http://bloomenergy.com/products/solid-oxide-fuel-cell-animat...
My last chemistry class was a long time ago, so this might be a dumb question. What's the difference between combining with oxygen and burning?
Fuel cells are cool because they grab electricity straight from the freed electrons as the fuel is oxidized. More efficient than using the heat to make steam to power a turbine to spin electromagnets, your typical power plant.
Fuel cells, for example, typically have a membrane at their core; that membrane tends to get clogged with crud. Stirling engines, on the other hand, are generally heavy (and thus expensive) and need to use exotic materials to handle high temperatures, which they need to operate efficiently.
Here is another article about them: http://www.fastcompany.com/1557348/bloombox-bloom-box-fuel-c...