When we determine the times of the festivals, we summon the holiness from above to shine upon us, and holiness pours forth upon the Jewish people.

We shall address this matter [of why the festivals are called “holy convocations”] by noticing that God commanded the human court to determine [the specific date of] the festivals, saying: “You shall proclaim them [tikre’u otam]” (Leviticus 23:2 inter alia); and [the spelling of otam is deficient;] it is written [as if it said] “You yourselves shall proclaim [tikre’u atem]” (Rosh HaShana 24a). By determining that a certain day is a festival, the court of the devoted of Israel imparts holiness to that day. As a result, [of the human court’s action,] the holiness above which is the root of each festival awakens and that holiness also establishes the day as a festival and pours out from above the abundance of holiness that is specific to that festival. Therefore, it is called “a holy convocation,” because on it, holiness from above is summoned, and the divine abundance of holiness descends to Israel below, and the day becomes holy. (Alsheikh, Torat Moshe, Leviticus 23)

Further reading: For the specific halakhot of each of the festivals see A Concise Guide to Halakha, p. 464.