Why does a wood fire create benzene?
This is actually just one part of a much more general phenomena, which is the release of large, polycyclic aromatic compounds due to pyrolysis or combustion of organic matter. It is not specific to wood. So, to understand why these compounds form in a wood fire, or any other kind of fire, we must first understand what wood is, and then we must understand how wood burns.

Wood is biomass - that is, matter composed primarily of carbons, hydrogens, and oxygens. We can generalize here, and talk about any biomass - wood, dead or dry grass, leaves, animal carcasses, fibers, sugar, fat, and even things like coal (which is fossilized biomass with special properties). All of these things contain cellulose or similar molecules, which consists of long chains of aromatic rings (the wikipedia article on cellulose is a good place to start, and any textbook on biochemistry will give much more detail). In the context of combustion, this is called "volatile organic matter." Volatile just means that if you heat it up, it burns, gives off heat, and turns into gas. 

Living things also have non-volatile and non-organic matter. As an example, people require some trace amounts of metals for biochemical functions, so humans contain some percentage of mineral matter. In the context of combustion, this material is often called "ash." If you have a fuel like wood, this ash is what's left behind after the wood burns, and composes much of what's released into the air in the form of smoke. Some fuels can have large percentages of ash (as an example, when organic matter fossilizes and forms coal, a large amount of rock or dirt interspersed with the organic material can lead to coals with very high ash content).

When you heat up organic matter, what you're doing is dumping a large amount of thermal energy into the molecules composing that matter, and after a certain point (depends on the fuel, but we're talking about 400 or 500 C) the large chains that compose organic matter begin to break apart. If this happens in the presence of oxygen, the material will also begin to oxidize, and the process is called combustion. If it happens in the absence of oxygen, the material will gasify. As the temperature gets hotter and hotter, a soup of big organic macromolecules is formed. (When I say big, I mean, BIG - hundreds of carbon, hydrogen, and oxygen atoms, and thousands of side chains and functional groups). It is in this macromolecular soup that you can find benzene and its many, many, many cousins. (These molecules are too complex to have names, and are so large that they can't be characterized exactly - instead, chemists can only statistically describe the molecules, using techniques like nuclear magnetic resonance (NMR), mass spectrometers (MBMS), or infrared spectrometers (FTIR).) This process of solid matter turning into volatile gases is called pyrolysis. 

Big polycyclic molecules form when there is not enough oxygen (if there were enough oxygen, all the carbon and hydrogen would form carbon dioxide and water). They are huge and hydrophobic, and agglomerate to form soot and tar. When you see sticky brown films on cigarettes or smokers' hands or in pipes, this is caused by these macromolecules.

If you looked very closely at a wood fire, and were able to see the actual flame-wood interface, what you would see is jets of hot, flammable gases shooting out of the pores of the wood. It is these gases that ignite - not the wood itself. (Of course, as the wood heats up, and the organic matter is burned, the mineral matter is left behind, and that becomes extremely hot but does not vaporize, so you still have radiant, red-hot material.)

You can still find benzene, even in a hot fire, because flames are not perfectly mixed, and you can get pockets of cold gas interspersed in a cloud of hot gas. If it is cold enough, benzene (which can be very stable) will not be decomposed.

So the next time you watch a campfire, or smoke a cigarette, or set your dinner on fire in the oven, you should be able to better understand what you're seeing - or know where to look.