What makes a wood burning stove clean burning?

Wood smoke is unburned fuel, some of which accumulates in your chimney as creosote while the remainder exits the stack as smoke. 
The key to reducing air pollution from woodstoves was to burn fuel more completely. Today's EPA certified wood burning stoves do exactly that.

Three things make today's stoves clean burning:
1.  How it is designed.
2.  How it is installed.
3.  How it is operated.

Many stove manufacturers use catalytic combustors to burn fuel more completely while others use a variety of design features such as baffles, secondary combustion chambers, and introduction of secondary air.

EPA-certified catalytic stoves offer up to 90 percent reduction of particulate matter over the older conventional model stoves. This is based on laboratory testing.