Every now and again Chazal use a method of exegesis of the form
אם אינו ענין ל___ תנהו לענין ___
which means that if a word or phrase is extra, due to the halacha it apparently teaches being already taught from another word or phrase, we learn a different halacha, which has less to do with the plain meaning of the text, from it. (Sefaria shows 63 places in Shas Bavli which apply this principle; some of these lines use it multiple times.)
This form of exegesis seems to take on one of two main forms:
1. Based on the second word or phrase, we're forced to interpret the first one differently. Ex. Shabbos 70a:
חילוק מלאכות מנלן אמר שמואל אמר קרא מחלליה מות יומת התורה רבתה מיתות הרבה על חילול אחד האי במזיד כתיב אם אינו ענין למזיד דכתיב כל העשה מלאכה יומת תנהו ענין לשוגג ומאי יומת יומת בממון
How do we know differentiation of melachos [i.e. that if one forgot it was Shabbos and performed multiple melachos he brings a Karban for each one]? Shmuel said, "The passuk says, 'Its desecrators will surely be put to death.' The Torah includes many deaths for one desecration." But this is written regarding an intentional violation! If its topic isn't for intentional violations, as it is written, "Whoever does melacha will be put to death," place its topic for unintentional violations. And what does 'he will be put to death' mean? He will "die" through money.
2. Based on the second word or phrase, we apply the first one to a different topic entirely. Sometimes we apply it to a specific topic, ex. Yevamos 95a:
אמר רבי אמי אמר ריש לקיש מאי טעמא דרבי יהודה דכתיב באש ישרפו אותו ואתהן וכי כל הבית כולו בשרפה אם אינו ענין לשרפה תנהו ענין לאיסורא
R' Ami said in the name of Reish Lakish: What is the reasoning of R' Yehudah [that if one is intimate with his mother-in-law, his wife is also forbidden to him]? As it is written, "In fire they will be burned - he and they." Should the entire house be burned?! [Only one of the women actually sinned!] If its topic isn't for the burning, place its topic for the prohibition.
Sometimes we don't apply this to any one specific topic, but rather all of them, ex. Kiddushin 43a:
גלי רחמנא בשחוטי חוץ דם יחשב לאיש ההוא דם שפך הוא ולא שלוחו אשכחן גבי שחוטי חוץ בכל התורה מנלן דיליף משחוטי חוץ אדיליף משחוטי חוץ ניליף מהנך הדר כתב רחמנא ונכרת האיש ההוא אם אינו ענין לגופו תנהו ענין לכל התורה כולה
[How do we know that there is no agency by sins?] The Torah revealed by slaughtering [offerings] outside [the Beis HaMikdash], "Blood will it be considered for that man, blood has he spilled." "[For] that [man]," and not for his agent. We have found this by slaughtering outside; how do we know by the rest of the Torah? We learn it from slaughtering outside. But the same way that we learn it from slaughtering outside, learn it from these [exceptions wherein there is agency by sins]! The Torah then writes, "That man will be cut off." If its topic is not for itself, place its topic for the entire Torah.
The way I understand this principle is that, at least exegetically speaking, there are no extra terms in the Torah. Thus, if we find a term that seems to be extra, we're forced to ascribe meaning to it.
What I don't understand is the second variation of this, wherein they don't even bother explaining how their new understanding of the text makes sense in context, nor how they know that their understanding is correct. Is there any rhyme or reason behind the application of this principle in these contexts?