The Vulgar Latin metipsimus, as a contraction of older metipsissimus, a superlative of metipse, comes from a reanalysis of egomet ipse "me myself" as ego metipse. In Latin, -met was attached to pronouns, but in Vulgar Latin it was reanalyzed as part of ipse instead.
Like Draconis stated, it's the weakening of the original word that caused it to be a demonstrative to fill the void that ille and iste left. It's similar to what happened in our language, "the" was originally a demonstrative as well in Old English, but weakened to merely a definite article totally in Middle English.
I believe the addition of the meaning "...
I will try to translate the reference, and hope for downvotes and corrections if I had anything wrong
Original from 2004-2008 Francesco Bonomi - Vocabolario Etimologico della Lingua Italiana
anche, anco sembra ad alcuni troncato
dal prov. ANCUI (a.fr encui, dial.lomb.
encoi) che significo quest'oggi, dal lat.
HANC-HODIE; altri lo tratto ...
Hanc hodie, literally "this today", is already attested in Plautus's time: the ho- element in hodie (originally a form of hic "this") had gotten semantically bleached until it was no longer emphatic, so an extra form of hic was added to get proper emphasis ("on this day!").
Loss of /h/, similarly, happened very early on: by the first century BCE we have ...