Dionysius of Halicarnassus in Roman Antiquites touches on, in brief, that the entiguishing of the flame was an omen for the destruction of the city.
There are many indications, it seems, when a priestess is not performing her holy functions with purity, but the principal one is the extinction of the fire, which the Romans dread above all misfortunes, looking upon it, from whatever cause it proceeds, as an omen that portends the destruction of the city; and they bring fire again into the temple with many supplicatory rites, concerning which I shall speak on the proper occasion. - Dionysius of H. Book 2, LCL 319, Pages 509
Livy also talks about the Sacred Fire of Vesta twice and speaks of it as an omen but again does not go into greater depth about WHY it needed to be alight and what would happen if it went out
Here is the Capitol where in the old days a human head was found, and this was declared to be an omen, for in that place would be fixed the head and supreme sovereign power of the world. Here it was that whilst the Capitol was being cleared with augural rites, Juventas and Terminus, to the great delight of your fathers, would not allow themselves to be moved. Here is the Fire of Vesta; here are the Shields sent down from heaven; here are all the gods, who, if you remain, will be gracious to you. - Livy 5.54.7
What need is there for me to speak about the perpetual fire of Vesta, and the Image —the pledge of our dominion — which is in the safe keeping of her temple? And you, Mars Gradivus, and you, Father Quirinus, what need to speak of your sacred shields? Is it your wish that all these holy things, coeval with the City, some of even greater antiquity, should be abandoned and left on unhallowed soil? - Livy 5.52.7
Cicero in Book 2 of De Legibus also says that the flame must be guarded but again does not touch on WHY
The several gods shall have their several priests, the gods all together their pontiffs, and the individual gods their flamens. The Vestal Virgins shall guard the eternal fire on the public hearth of the city. - Cic. De Legibus 2.20
Plutarch, Pliny and others, talk to great lengths about the punishments Vestals received for letting the fire go out. However, apart from the excerpts above, I cannot find any other gobbets about WHY the fire needed to stay alight.
Could anyone help me out with any further ancient sources that talk about this? Thank you!