The dates come from the Life of Vergil (the Vita Vergiliana) by Aelius Donatus, which is mostly based on Suetonius' lost On poets (De poetis). The second sentence of the Vita gives us the birth date as follows:

natus est Cn. Pompeio Magno M. Licinio Crasso primum coss. iduum Octobrium die in pago qui Andes dicitur et abest a Mantua non procul.

The death notice is given further down as follows:

...languorem nactus est eumque non intermissa navigatione auxit ita ut gravior aliquanto Brundisium appelleret, ubi diebus paucis obiit XI Kal. Octobr. Cn. Sentio Q. Lucretio coss.

The standard edition of Donatus' Life is in

• Brummer, Iacobus (ed.), Vitae Vergilianae (Leipzig: Teubner, 1912)
• Brugnoli, Giorgio, and Fabio Stok (eds.) Vitae vergilianae antiquae. Scriptores graeci et latini. (Rome: Istituto Polygraphico, 1997.)

but I'm afraid I don't have access to a copy to get exact page numbers.

Update for the OP: I'm posting this separately in case you don't check back. There's a new standard edition of Donatus dating to 1997, which is

• Brugnoli, Giorgio, and Fabio Stok (eds.) Vitae vergilianae antiquae. Scriptores graeci et latini. (Rome: Istituto Polygraphico, 1997.)

As a point of interest, I've now seen that the Life that was most commonly treated as the main source up to the 19th century, known as "Donatus auctus", gives the death date as the 22nd of September as opposed to the 21st. There's a text here. Donatus should be taken as authoritative, but there's always a chance of an error in the online text I linked you to. Either of the critical editions should make the situation clearer.

Edit. Though the New Pauly gives the same death-date that I quoted from Donatus (21st of September), so the text does appear to be accurate. It's just Donatus auctus that's wrong.