I think that the birds represent the idea of a threatening presence that is larger than that of the human being. These threats are the ones that threaten our basic existence and we, as human beings, are unable to stop what is there. Nat cannot stop the birds. In the end, the best he can do is to be mindful of their presence, not deny it, and then seek to protect his family at every waking moment. In this light, the birds can be seen as those unavoidable forces that shadow and loom over our consciousness as human beings. The presence of these forces seem to compel individuals to pay attention and be mindful of their external or objective world. Interestingly enough, though, the response to this does not seem to be to further engage in the external world, but rather retreat to the internal. In this light, the birds function as the catalyst for individuals to retreat to their internal domain in the face of overwhelming objective odds. The Cold War historical context of the work helps to bring this out in even more force in sensing the importance and the function of the birds.
Many have argued over the years that the birds represent the power of nature and mankind's stubborn inability to recognize the power of nature.
Throughout the story, the characters are at the mercy not only of the birds, but also of the wind and sea. Everything seems to work in concert to attack this town. Of course, the characters try to control the situation for their own safety, but they are simply outnumbered and overpowered.
Humans are at the top of the food chain, but that often goes to our heads and we fall into the belief that we can control nature. It is not usually until some natural disaster that we are reminded that we live at nature's will. One landslide, earthquake, wildfire, or in this case, bird swarm, can throw our entire world into chaos. This story reminds readers of mankind's hubris and suggests that even small and normally non-threatening things can become a threat when they act together.