I feel like your question would be better answered by philology experts than historians. However, the transmission of texts is a matter on which our disciplines overlap. That's why I allow myself to give it a go.

How were texts passed on to new generations before the printing press?

The printing press really secured the written text. You can have the exact same hard copy of a text in several books. This was unheard of before the 15th century in Europe.

If we look at manuscript traditions of "popular" texts from the 13th to the 15th century, we observe a lot of freedom was taken in regard of their original version. More often than not, of course, texts were copied in their entirety. However, it was common practice to change their apparatus. Chapter titles would be skipped, changed or added. Sometimes, only a selection of chapters would be transcribed. They could even be enriched by chapters from another text when they were not included in some "best of". Texts were not "set in stone" and could be integrated in many various contexts.

I would shortly present the MS Ll.1.18 (Cambridge University Library) to make my point before moving on. As Richard Beadle recently stated, this manuscript is a 15th century miscellany. It contains extracts of various origin. Many passages, however, are fragments of John of Burgundy's plague tract. This show that the main scribe writing this manuscript was quite concerned with information related to the plague--pretty much like we're obsessing right now about the Covid-19. As a matter of fact, plague returned in England in 1485, around the time the manuscript was done. Fragments of John of Burgundy's plague tract therefore naturally found their way in the miscellany. That's because not all manuscripts were made to be "proper editions" of classical, historical or literary texts. They could fill other purposes.

However, some scribes had the specific duty to copy as exactly as they could texts of the utmost cultural impact. There was no dabbling in transcribing the Bible or Augustine's works, for example!

If the text was written in a vernacular language, a modernization of its phrasing was expected. The best copy that we have of Joinville's biography of Louis IX of France is dated from the late 14th century. It's quite problematic because Joinville died in 1317. That manuscript also displays an orthography that Joinville would never have used. We know quite well how he wrote for he left us many charters. However, around the reign of Charles V, the French language went through a lot of changes in regard of its written form. Clerics wanted to get it closer to Latin and they started to write down letters in the middle of words that were not pronounced but acted as reminders of Latin etymology. That's why we have a 'g' and a 't' in "doigt" (finger) though we never pronounce it, because it traces the word back to its Latin etymology: "digitus." Joinville would have been foreign to such concerns. The manuscript that contains the best copy of his biography of Louis IX is therefore "tainted" linguistically. However, there is much to wager that the actual content of the book didn't change for one bit because the text, in and of itself, was deemed as too important to be altered.

When the text was written in classical Latin or in ancient Greek, now, the scribe had better not to change anything from it since it was already destined to scholars, intellectuals and clerics who had studied the antique languages. Such people were also hardpressed on insuring those language didn't evolve but remained in their "pure" form. They'd read and studied Virgil, not only for the content of his works, but mostly for his grammar and rhetoric prowess! The same was true for Byzantine scholars and their study of ancient Greek texts.

What is a textual tradition?

When texts used to be copied over and over before the printing press, from scribe to scribe, we talk about textual tradition (or transmission). In the introduction of any serious edition of an ancient text, you'll find a few pages about its textual tradition. Meaning an extensive study of the manuscripts that contain the text in question and how they relate to each other. You can even have entire books dedicated to the study of a textual tradition. It wouldn't be surprising for any classical text since a few of them are found in several hundred manuscripts if they ever became popular among clericsduring the Middle Ages. Ovid's Metamorphosis has been found in no less than 489 manuscripts according to the latests results of El Grupo de Investigación Nicolaus Heinsius based in Huelva (Spain) and we could still find more of them!!!

Once philologists have tracked down as many manuscripts as they can that contain the text they're interested in, they start to study the many variants in those manuscripts. I mean that they look at every. single. word. and spot all. the. differences. they can. Even the tiniest difference make the cut! Nothing is being left out. Every word on every page is scrutinized. That's when patterns start to show up. If we find out that several manuscripts share the exact same variant, we'd argue that belong to the same "family" so to speak. They all stem from the same hypothetical or physical model. Then, once the manuscripts are sorted out in such a fashion, philologists wonder which manuscript would be the "best"? Which manuscript would display a text that is the closest to its original rendition?

Context, at this stage, becomes key. We may not have the original copies of Virgil, Homer or Ovid, but we have other sources of information when it comes to study ancient languages. Once you've compared all the manuscripts containing several texts dated from the same era and which are supposed to share common linguistic traits, in order to build up a proper grammar of the language in which they're written, you can confront your results to texts that actually survived the passing time. Indeed, we still have written texts that physically date back from the Classical Era, either on stone or other materials. This is another field of study that we call epigraphy, or the study of ancient inscriptions. Either on tombs, official monuments, houses, temples, or other, we found several thousands of ancient inscriptions of the highest linguistic value. The Rosetta Stone comes to mind... Hieroglyphs would still be a mystery without it!

At the end of all those extensive studies, we can firmly define which medieval manuscript shows the "best" version of a text depending on the historical context of its author.

How we know what we know about ancient texts and go about looking at them Is a pretty interesting topic and I think your question can be more or less be summarised as this

• We generally don't have original texts. Why is this? Does this create problems and how do we know that the versions we have are accurate and haven't been tampered with?

I wrote this a little while ago. It goes into how texts have been passed down to us and the process of reconstructing them. I think that should answer your question?

The brief summary is that we have copies of copies of copies of original texts. Errors creep in over time in these copies for various reasons. It's a pretty long and extensive process to try and weed out these errors to the best of our ability to create what we believe is a version most accurate to the original. But ultimately it has some imprecision. We can never know for certain that we're right, everyone is just doing the best that they can.

Your question is fairly non specific and doesn't need any detail specific to Herodotus I think, so I think/hope that more generalised answer I gave should be helpful and answer your title too.

You might enjoy reading this answer about textual criticism by /u/LegalAction