The etymon is present only in South Slavic. Its original meaning probably was to make a connection, which eventually evolved into to recall a memory > to remember. Despite the superficial resemblance, the current etynom is distinct from *(u)sętiti(“to sense”).
From Proto-Indo-European*swēt-. Cognate with Lithuaniansvẽčias(“guest”), Ancient Greekἕταρος(hétaros, “comrade”). Per Derksen, the lengthened grade of the root is unexpected. The Indo-European root comes from the more basic root *swé(“self”) with a root extension.
The semantic shift relative, kin > to host is similar in nature to the development of Germanic *fulgāną(“to follow”) from *fulką(“people”). In both cases, the original meaning likely was to establish good relation with your relatives, which eventually evolved into to host or to follow, respectively.
Chernykh, P. Ja. (1993) “посети́ть”, in Историко-этимологический словарь русского языка [Historical-Etymological Dictionary of the Russian Language] (in Russian), 3rd edition, volume 2 (панцирь – ящур), Moscow: Russian Lang., →ISBN, page 60
Derksen, Rick (2008) “*posětiti”, in Etymological Dictionary of the Slavic Inherited Lexicon (Leiden Indo-European Etymological Dictionary Series; 4), Leiden, Boston: Brill, →ISBN, page 414
Vasmer, Max (1964–1973) “посети́ть”, in Oleg Trubachyov, transl., Этимологический словарь русского языка [Etymological Dictionary of the Russian Language] (in Russian), Moscow: Progress