The three prayer services that we recite each day correspond to the three patriarchs as well as to the daily sacrifices in the Temple. Shaḥarit and Minḥa correspond to the two daily (tamid) offerings in the morning and afternoon, respectively, and Ma’ariv corresponds to the nighttime burning of fats and limbs on the altar (Berakhot 26b; Peninei Halakha: Prayer 1:7). The Sages added the Musaf prayer – to be recited on Shabbat, Yom Tov, Ḥol Ha-mo’ed, and Rosh Ḥodesh – corresponding to the additional (musaf) offerings sacrificed then.
The Musaf Amida of Yom Tov (except Rosh Ha-shana) contains seven berakhot. The first three are similar to those of every Amida, while the middle berakha relates to the festival offerings. We begin by stating that on account of our sins we were exiled from our Land, and our Temple was destroyed, so we are unable to bring sacrifices as we once did. We then plead:
Bring back our scattered ones from among the nations…. Lead us to Zion, Your city, in jubilation, and to Jerusalem, home of Your Temple, with everlasting joy. There we will prepare for You our obligatory offerings; the regular daily offerings in their order and the additional offerings according to their law.
We then mention the name of the festival. Ashkenazim also recite verses pertinent to the musaf sacrifice. We go on to pray that the Temple be rebuilt and that we be privileged to fulfill the mitzva of making a pilgrimage to the Temple three times a year for the festivals. We conclude with the prayer of Ve-hasi’enu, as we do at every Yom Tov Amida (section 3 above).
Following Ashkenazic custom, at the conclusion of the Retzei section about restoring the Temple service, the ḥazan adds the formulation that was said in the Temple: “May our entreaty be as pleasing to You as a burnt offering and sacrifice. Please, Compassionate One, in Your abounding mercy restore Your Presence to Zion, Your city, and the order of the Temple service to Jerusalem.” The ḥazan concludes: “Blessed are You, Lord, for You alone do we serve with reverence” (She-otkha levadkha be-yir’a na’avod”). Some follow the Vilna Gaon’s practice and instead conclude the berakha in the usual way: “Blessed are You, Lord, Who restores His presence to Zion.” If no Kohanim are present, then Ve-te’erav is omitted (MB 128:173).