I'm very curious to understand how we got texts from classical antiquity. The story from the middle ages generally seems to be that they were rediscovered in a religious institution, copied and circulated around Europe from the 14th century, helping the renaissance to flourish. But what were these texts doing for the preceding millennium? And how did they get onto those dusty shelves to begin with?
Take as an example Caesar's Commentaries, copies of this would have been scribed in Rome in ~50BCE, (on what material?), it would have kicked around rich senators private libraries for a few hundred years, what is the link from the fall of the WRE to its rediscovery? Of course every text will have its own story, but what are common stories?
(Not talking about fragmentary texts on clay or archaeological work here, but the longer texts that have survived from Greek and Roman antiquity).
You've got a good set of links below that discuss the manuscript tradition. I'm just going to give you a note on Caesar specifically, since that's the example you chose.
We know nothing about the publication of the Commentarii. We don't know when they were written (obviously around the wars - more on that shortly), or how they were published, or who the intended audience was.
We know Cicero read them, and considered them commentarii, because he wrote letters about them. Commentarii are meant to be source material for a proper historian to use while writing a proper literary history. Cicero was so impressed by Caesar's writing he thought no one should try to use them that way.
A very popular, if undocumented, theory is Caesar composed these commentarii during the winters, when fighting wasn't expected. But Caesar is known to have been a prolific writer of letters and there's no reason he couldn't write these on campaign. We just have no evidence about it.
It is also popular to think (again without evidence) that the commentarii were a sort of status update sent to the senate. If that were true, Caesar would be the only known person to practice such a thing. On the regular, I mean. It's not entirely clear what the purpose of such status updates would serve. I'm on a two night insomnia streak, so may not be firing on all cylinders, but the only senator I can think of who is on record caring anything about what Caesar was doing in Gaul is Cato. For the Civil War books, it's even stranger, since half the senate was openly hostile to Caesar and hanging out in Greece.
The other possibility is the commentarii were presented at contiones. Those are public meetings used, among other things, to present bills the people would later vote on. This is where they would get their information about government for the most part (Caesar did publish the official acts of the senate when he was consul, but that wasn't standard or traditional).
In both cases, the commentarii must have been read out loud. No one is copying out 600 individual copies on papyrus so each senator can have one, nevermind the contiones. How Cicero got his hands on a copy, I have no idea. Cicero and Caesar were... on polite terms, if not exactly friendly. I suppose it's possible Caesar sent him a draft. Cicero was known for his own literature, and we know he sent drafts of his stuff to Atticus, and while that's not exactly analogous, because Atticus was Cicero's publisher, it's not outside possibility that Caesar did the same with Cicero.
So we have no evidence about when or in what circumstances (on campaign or in the winter) these were written, for whom they were written, how they were published, and what their purpose was.
If I understand your question correctly, you're asking about the transmission of classical texts during the middle ages. The general answer is what's called the manuscript tradition; that is, in addition to reading and studying and writing about classical texts, medieval scholars (mostly in religious institutions) were continually making copies of these texts. There are a lot of good explanations of this topic on r/Askhistorians, including but not limited to:
/u/kiwihellenist on Tacitus' histories and on Caesar's commentaries
/u/legalaction on the works of Cicero
/u/tinyblondeduckling on Augustine of Hippo's Confessions