I don't really know what is the link between bird and the brain.
The only conclusion that i can make out of the saying is, bird is a creature with small head, thus making them having small brain.
For example birds like swallow, robins and tweety bird. But mind you, tweety is quite brainy because she can always get away from the naughty cat.
Somehow, i pity all the birds families. Because of one tiny bird with slow-mo brain, human, being ever judgemental as they are, came out and suggest that birds are all stupid. Nevertheless, i dont really know whether is it really the size of the brain that makes one smart or otherwise.
Brains on the Web: Comparative Neuroanatomy and Intelligence