This is an example of what's called conjecture. You can read a basic introduction to it on Wikipedia:
Conjecture (conjectural emendation) is a critical reconstruction of the original reading of a clearly corrupt, contaminated, nonsensical or illegible textual fragment. Conjecture is one of the techniques of textual criticism used by philologists while commenting on or preparing editions of manuscripts (e.g. biblical or other ancient texts usually transmitted in medieval copies). Conjecture is far from being just an educated guess and it takes an experienced expert with a broad knowledge of the author of the text, period, language and style of the time. Conjecture requires a close study of the text in its cultural and historical context and must be preceded with a thorough analysis of all extant versions and readings of the given fragment. The knowledge of writing styles used by the scribes throughout the transmission stages is also essential. Conjectural emendation must be clearly indicated in the critical apparatus or in the text itself.
The basic idea is that scholars, when they read a text and find something that doesn't make sense, believe that the text itself is corrupt. This is common enough. Every single manuscript contains at least one error, and some more than others. The job of the textual critic is to figure out what the original text looked like. So when we have mistakes in the text (grammatical, spelling, missing words, etc.), textual critics need to figure out why.
When a scholar changes the text as we have it in any manuscript, we call that "emendation" or "conjecture" (the two words are synonymous or nearly synonymous, with the differences being a technical matter). The idea is that they have an idea (a conjecture) about how the text should actually look like. So they emend (=fix) the text back to (in their estimation) what the original would have looked like.
Sometimes this is simple enough. Due to scribal practices, words were often left out of copying. We even have ancient quotes of a text that differ from what the manuscripts say.
Haplography is also another problem: this is skipping words or lines because the eyes see a similar set of letters further on than the word they are writing and inadvertently skip over words in the process.
As an English example, think of the sentence:
I went to buy a cat from a catholic church. It was cute and fuzzy.
In a manuscript, you might find:
I went to buy a catholic church. It was cute and fuzzy.
Now, that makes no sense, especially when the next with the second sentence. So a scholar might conjecture either the original sentence (if they're lucky!) or a sentence like:
I went to buy a cat [from a pet store near a] catholic church.
The brackets would indicate that the editor has emended the text and believes that the text as we have it is corrupt and this is an adequate fix.
With prose, this is difficult, but in poetry it's a bit easier due to meter. Sometimes we'll see a line and notice that it doesn't scan. So editors try to emend the line so that it scans and makes sense.
Can all this change the meaning of the text? Well, it changes the meaning of the text as it is in the manuscripts. Since we have no original manuscripts, the whole enterprise is tenuous at times, and new discoveries (as rare as those are these days) might invalidate a lot of ideas. Sometimes scholarship will also call into question the need to emend a text at all. I'm thinking of Marilyn Skinner's article "Ego Mulier", in which she argues that the words ego mulier ("I, a woman") in Catullus 63.63 does not need emendation, even though Catullus is not a woman, because of, well, quite a few reasons, but basically that this is poetry, and he doesn't have to describe himself accurately in poetry, and he might not be describing himself anyway, and even if he were, gender boundaries were looser then than they are now, etc.
Things get even stickier when the Bible is involved, due to religious beliefs. Just like any ancient text, all biblical manuscripts have errors in them, and sure enough, sometimes emendations have provoked ardent defenders of the faith; other times it has created new faiths.
All this is to say that conjecture is an enormous topic and there's no easy way to provide just a few examples. But if you want to learn more on the topic from a scholarly point of view, try Martin West's Textual Criticism and Editorial Technique Applicable to Greek and Latin Texts or more recently Richard Tarrant's 2016 monograph Texts, Editors, and Readers.