Look up "phrase-final vowel devoicing" for scientific articles on the subject.
It's a relatively recent phenomenon in European French, whereby the vocal folds stop vibrating halfway through a vowel at the end of an utterance. Since the tongue is still articulating the vowel and air continues streaming out of the mouth, this produce a fricative. As you've noticed, this produce an ich-Laut after a high vowel, but I make ach-Lauts after low vowels too sometimes ("Il n'y en a pas" --> [jɒ̃napax])
It's totally allophonic, meaning native speakers typically won't be aware they (and the people around them) are producing these kind of fricative themselves, unless they pay a lot attention to phonology and pronunciation. They might deny that it happens at all.
It also appears to be associated with newscaster's speech, but is far from limited to them. Rather, it's a phenomenon that's more prevalent in careful speech and started as a prestige variant. It has spread far enough by now that some speakers are beginning to be aware of it and express negative sentiments towards it ("relachement vocalique"), which is indicative of an ongoing sound chance.
As for where it happens, my sources tend to describe it as a general phenomenon. I know I've heard it from French folks and younger Belgians.
Every oral vowel can potentially be devoiced, but it's more common by far with /i/ and /y/. My sources disagree as to the percentage of devoicing for each vowel so I'm not going to attempt to summarize them.
Devoicing typically happens at the end of a declarative sentence, with falling tone. It's used as a way to signal the end of the speaker's speech turn. That means that a simple "merci" will be pronounced [mɛχsiç], but "merci à votre fils" [mɛχsi a vɔt fis:] without devoicing.
That said, there are a few reported cases in the literature of devoicing of "oui" in the middle of an intonation group. High-frequency words also have their final vowel devoiced more often than rarer words. This could indicate an ongoing lexicalisation of the epithetic fricative, that would then appears in every context. But we're not there yet.
Finally, it could be related to other end-of-phrase signals in French, such as lengthening of final fricatives (my [fis:] above) or a strong aspiration of phrase-final unvoiced stops. These also function to close the speaker's intervention in a conversation and signal to their interlocutor they can start speaking.
Un vrai ami, by cilka (compare with gwen_bzh's pronunciation without devoicing)
Moi aussi, by Lexou (again, constrast with gwen_bzh)
[yçʷ] (you might hear hear /f/ because of the lip rounding of the vowel)
Armée du Salut, by ecureuil
[uçʷ] (same remark)
Je t'aime beaucoup, by tibboh and (especially) Claire19
-Candea, Maria. "Au journal de RFI-chhh et dans d'autres émissions radiodiffusée-chhhs. Les épithèses consonantiques fricatives." Le discours et la langue 2.2 (2012): 136-149.
-Dalola, Amanda. "The role of vowel type, preceding consonant and lexical frequency on final vowel devoicing in Continental French"
-Paternostro, Roberto. "Le dévoisement des voyelles finales." Rassegna italiana di Linguistica applicata 3.40 (2008): 129-158.
-Smith, Caroline L. "Vowel devoicing in contemporary French." Journal of French Language Studies 13.02 (2003): 177-194.