Augustin Goderniaux, you are definitely NOT hallucinating. You are experiencing the so-called "intrusive R" phenomenon. Intrusive "R" occurs when speakers want to avoid two vowel sounds occurring next to each other and acts as a connector, between a word-final schwa /ə/ spelt with an < a > (e.g. China, Lisa, America) and a new word beginning with a vowel sound. Ex: Lisa –r– Edwards. In some ways it’s a strange phenomenon because native speakers often don’t realise that they’re doing it! Note, that the intrusive "R" is not spelled. Don't confuse it with a "linking R", which is spelled, like in: care about,

I think the purpose is to properly distinguish between words. Without emphasising the end of "pizza", "pizza" and "out" could blur at the vowel sound and almost become one word lol. It's easier to add the emphasis than it is to pause and make sure you stop one vowel before starting the next. So, "pizza (pause) out" can almost become "pizza route". :P

The intrusive-r is a characteristic of British English. Briefly, British English tends to drop the r at the end of syllables (card->cawd) and add an extra r between some words. North American English tends to pronounce the r in all positions and not to add an extra r between words.

I thought that was called "Connected Speech" > Linking 'r' but it turned out that it's actually "Intrusive R"

From Wikipedia - linking R and intrusive R https://en.wikipedia.org/wiki/Linking_and_intrusive_R Linking R In many non-rhotic accents, words historically ending in /r/ (as evidenced by an ⟨r⟩ in the spelling) may be pronounced with [r] when they are closely followed by another morpheme beginning with a vowel sound. So tuner amp may be pronounced [ˈtjuːnər æmp].[nb 1] This is the case in such accents even though tuner would not otherwise be pronounced with an [r]. Here, "closely" means the following word must be in the same prosodic unit (that is, not separated by a pausa). This phenomenon is known as linking R. Not all non-rhotic varieties feature linking R. A notable non-rhotic accent that does not have linking R is Southern American English.[9] Intrusive R The phenomenon of intrusive R is an overgeneralizing reinterpretation[10][11] of linking R into an r-insertion rule that affects any word that ends in the non-high vowels /ə/, /ɪə/, /ɑː/, or /ɔː/;[12] when such a word is closely followed by another word beginning in a vowel sound, an [r] is inserted between them, even when no final /r/ was historically present.[13] For example, the phrase bacteria in it would be pronounced /bækˈtɪəriərˌɪnɪt/. The epenthetic [r] can be inserted to prevent hiatus, two consecutive vowel sounds.[14]