The Church commemorates the saints on their dies natale (Latin: “day of birth”). This “heavenly birthday” is not the day the saint was born, but the day the saint received the crown of life (Jas 1.12; Rev 2.10). For most of the early saints, this dies natale was their day of martyrdom. For this reason, the dies natale supersedes the day they entered this world in the flesh. Hence, unlike the world which celebrates its hero’s birthdays, the Church celebrates her hero’s death-days—and calls them “true birthdays” (dies natale).
There are, however, three notable exceptions to this rule: the Nativity of Our Lord (25 December), the Nativity of the Blessed Virgin Mary (8 September) and the Nativity of St John the Baptist (24 June). One reason is that these are the birthdays of the only individuals who were without, or cleansed from, ancestral sin. St Augustine suggests another reason: “The Lord wished John to be an attestation to his own first coming; for if Christ had come too suddenly and unexpectedly, men might not have recognized him. And on this wise John was a figure of the Old Testament, and showed in his own person a typical embodiment of the Law; for he heralded beforehand the coming of the Savior, even as the Law was our schoolmaster to bring us to the grace of Christ.” In other words, St John’s nativity points to Our Lord’s nativity; which means that the Nativity of St John the Baptist teaches us (as do all the feast of the Blessed Virgin Mary) that Our Lord God deigned to take our human nature so that He might dignify and elevate it by penetrating it with His divine nature.
For more by St Augustine on this feast, see this excerpt.
For an fine piece on the same topic by St John the Wonderworker of Shanghai & San Francisco, see this piece at Christ is in our Midst.