I don't understand where vosmet and temet came from. I know vos and te as pronouns, but what is the -met ending? Is that from some other language? Is it used anywhere else? It seems irregular. Why even use these words instead of the regular form for yourself (ipsum)?
It's for emphasis, and older than the use of ipse as an intensifier. From Allen & Greenough §143.d:
Emphatic forms of tu are tute and tutemet (tutimet). The other cases of the personal pronouns, excepting the genitive plural, are made emphatic by adding -met: as, egomet, vosmet.
NOTE.—Early emphatic forms are mepte and tepte.
Wiktionary has a list of them all.
You can stack them, too: egomet ipse.