Air alone does not burn, but wood smoke
, which is largely a product of incomplete combustion
(usually due to lack of enough oxygen to support complete
contain combustibles which will
burn when provided oxygen
in the presence of enough heat
(the wood smoke is the fuel
in this example of the classic fire triangle).
That's what's happening at the very hot region where additional air is introduced near top of these 'stoves', resulting in reduction of the smoke to approaching nil. Yes, there are likely still small particulate and fume fractions of the smoke which are not combustible in these conditions and may escape
The "contraption" simply provides a more efficient setting for all of this to occur than the typical 'open fire ring', by confining and directing the heat, oxygen, and fuel to a region where they can all work to the result of more complete combustion.
It's not voodoo or a sham, just a reasonably elegant application of combustion science.