Mr. Webb's comment about the townspeople paying special attention to the birds indicates at least two things about Grover's Corners.
First, life in Grover's Corners is quiet and without event or excitement, for the most part. Life in this little town is calm. It is so calm that the birds become an attraction of interest.
The notion of the town's subdued and rural nature connects it with an era that has passed by the end of the play (when cars are showing up on the streets with increasing frequency).
The other idea signified in Webb's comment relates to the theme of nature, natural order, and the cycle of life. The play is concerned with portraying and commenting on the normal and universal patterns of American life at (or just after) the turn of the century.
Our Town is concerned with the great and continuing cycle of life; out of life comes death and from death comes life.
We see this in the dialogue relating to the universality of marriage and in other comments made by the characters (Mrs. Soames; the dead, etc.).
The birds are a symbol of nature, connecting with the play's interest in exploring the natural cycle and pattern of life.
This cycle is man's closest understanding of eternity, his finest artistic expression of what he senses to be a mission and a purpose.