Why the birds? Some see the birds as symbolic of Satan, since the birds are "the evil one" in Jesus' first parable (Matt 13:19). However, symbols don't mean the same thing every time they occur. In this case, Jesus is drawing on an Old Testament image to enrich his parable. The picture of a great tree that provides a haven for birds is used three times in the OT (Ezek 17:23, 31:6; Dan 4:12). In each case, the tree is a mighty kingdom and the birds depict the majesty and blessings of that kingdom.
At first look, Jesus could be alluding to any one of these three OT passages. His wording does not perfectly match any of the three passages, which is quite normal when the NT refers to the OT. A closer look shows us that Jesus is probably echoing Ezekiel 17, which contains a provocative parable about the fall of the royal family of Israel in 586 BC. The family is a vine, which is uprooted and withers because of its unfaithfulness to its gardener. But one day, God will take a tiny twig, plant it in Jerusalem, and it will become a great cedar, "and birds of every kind will nest... in the shade of its branches."
Many Jews at the time of Jesus believed that Ezekiel 17:23 described the rule of the messianic king, ruling over a restored Jewish kingdom. The translators of the LXX (the Greek Old Testament) called the cedar "he" to make it clear that the cedar is a person (in Greek, cedar is a feminine word); and the Ezekiel Targum (an Aramaic translation, possibly from the late first century) explained that the cedar was a king from the line of David, and the birds were the "humble who dwell in the shade of his kingdom."
So Jesus ends his parable with an allusion to Ezekiel's older parable. Ezekiel's tiny twig would become a mighty cedar. Jesus' mustard seed would become a great tree. The dusty rabbi from Nazareth and his nondescript band of disciples would become a mighty kingdom, providing blessings to all who recognize the majesty of the king and his kingdom.
The picture: Jesus Teaches His Disciples, in Das Plenarium oder Ewangely Buoch, printed 1516. Courtesy of the Digital Image Archive, Pitts Theology Library, Candler School of Theology, Emory University.