How do they do it?

[bullpen]

How do birds avoid hitting trees when they fly through woods?

[burnhal]

Probably a lot of factors involved.

They know their area well.
Their view of the world is not the same as ours- we are a lot bigger and see a bunch of tangled foliage as a barrier- they will see paths through it.
Their movements/reactions are a heck of a lot faster than ours. Their eyesight is a heck of a lot better as well.

[wildwood]

In addition to burnhal's excellent answer, birds also have a very sensitive nerve layer under the skin just as we have. They can feel the tiniest change in air pressure on their feathers and with their lightning fast reaction they can change direction very quickly. This ability also enables them to fly and circle in large flocks in what appears to be total unison.

[plowter]

Could it also be due to evolution?

Birds which were crap fliers and kept bumping into trees wouldn't last very long. So those individuals in each species which did have the spatial orientation thingumy and good reactions would survive and pass their genes (and experience) to their offspring.

Of course, creationists may dismiss this explanation since God would only make birds that wouldn't bump into trees.