Wow! I had no idea...
The mesh goes in just above the aquarium pump air inlet at the bottom of the mug, so that the air is free to circulate through the wood chips/chunks. The principle is like continually blowing across a smoldering piece of wood.
Some smoking processes involve burning wood in the absence of oxygen (this is what you are doing with a smoker box, or by wrapping wood chips/chunks in foil...you are restricting the amount of oxygen supplied to the burning wood, as well as protecting it from an open flame).
Wood burned in the absence of oxygen creates charcoal, which is why so many of you have noticed that wood consumed that way resembles lump charcoal. On the other hand, wood burned with an adequate supply of oxygen is consumed completely and leaves behind only ash. The idea is to control the amount of air flow so that the wood merely smolders and does not ignite into an open flame. What you have left should be a small amount of ash.
Additionally, the concept of burning wood in an adequately supplied oxygen environment is that, by controlling the amount of air, you can control the amount of smoke...ie, the more air supplied, the more smoke (and heat) generated, the less air, the less smoke (and heat).
Controlling the smoke independently from controlling the heat...now that offers some tantalizing possibilities.
Other considerations are that the wood chunks to wood chips ratio needs to be adjusted so that the smoker becomes self-sustaining, and this depends on the size of the container. Bigger containers can feed bigger chunks and fewer chips, and conversely, smaller containers need more chips and smaller chunks.