The phrase in the question is drawn directly from the Gemara's description of the World to Come, and Rambam (Hilchot Teshuvah 8:2) cites it verbatim — that there the righteous sit with their crowns on their heads and enjoy the radiance of the Shechinah — explaining that 'sitting' is a metaphor for being present without toil, and the 'crowns' represent the knowledge of what earned them this reward.
Rambam (Hilchot Teshuvah 8:1) grounds the concept of 'a day that is wholly long' and 'wholly good' in a drashah on the pasuk of long life and goodness in this world (Devarim 22:7 as read there), identifying these as code for the World to Come, which is a world entirely good and entirely long.
Rashi (Berakhot 17a) explains the companion phrase that the righteous 'ate and drank' before God to mean that they were sated from the radiance of the Shechinah as though they had actually eaten and drunk, and Sha'ar HaGemul of the Ramban (60) develops this at length, citing the same Gemara in Berakhot and comparing the sustenance of the righteous by the Shechinah's radiance to the way the soul is sustained by the body in this world, bringing Moshe Rabbeinu as a precedent for being nourished by that radiance.
Tanna DeBei Eliyahu Rabbah (4) links this state to the eternal Shabbat, teaching that the World to Come is called the day that is wholly Shabbat, a world without death, sin, or punishment, in which each person rejoices in his own wisdom and understanding.
Chochma uMusar (1, p. 287) offers a reading of 'their crowns upon their heads,' explaining in the name of the Baal HaTurim that the crowns are the very crowns that Israel gave to God through their prayers three times daily throughout their lives, which God will return to the righteous in the World to Come.
Mishnah Sanhedrin 10:1 establishes the broader principle that all of Israel has a share in the World to Come, citing the pasuk that God's people shall all be righteous and inherit the land forever, while also listing those who forfeit that share, such as one who denies resurrection or the divinity of the Torah.
Daniel 12:2, Yeshayahu 25:8, Tehillim 16:11, and Yeshayahu 64:3 together supply the scriptural background for the themes invoked in the prayer — the awakening of those who sleep in the dust to eternal life, the destruction of death and the wiping away of tears, the fullness of joy in God's presence, and the unseen good prepared for those who wait for Him.
Rambam (Hilchot Teshuvah 8:3) adds that this life is called 'the bond of life' precisely because death, which is a bodily event, has no place there, since the World to Come has no body — and this reward is one than which there is no greater, the good after which there is no further good, the very thing all the prophets yearned for.
Abarbanel reads Daniel's prophecy of those who awaken to eternal life as referring to the righteous who died in exile, who will be revived, enjoy the Leviathan, die a second time, and then be revived again at the resurrection to the World to Come, a world without eating or drinking but only delight from the radiance of the Shechinah.
Sefer HaIkkarim (Maamar 4:31) distinguishes a broad and narrow sense of 'World to Come': broadly, any degree of reward the soul attains after death, available even to the righteous of the nations; narrowly, the supreme degree attained only by the fully righteous after resurrection, which is called 'life in the World to Come' in the specific sense invoked by the prayer.
Torat HaAdam, Gate of Recompense affirms that the consensus view maintains resurrection and the preservation of the people who are resurrected forever, from the time of resurrection onward, into the World to Come which is 'a world that is wholly long,' apart from the Rambam's own more limited position on the duration of the resurrected state.
Recanati (Achrei Mot:27) and Sefer Chasidim (604) both describe the World to Come as the place of ultimate recompense held in reserve after death — Recanati describing the soul's sojourn facing the Shechinah until the resurrection and the final apportioning of the World to Come, and Sefer Chasidim, citing Rav Nissim Gaon, describing this world as the place of deeds and the World to Come as the place of payment, where the righteous take nothing of their reward in this world so that it remains stored for them there.
Tanya (Likkutei Amarim 36) and Nefesh HaChayim (Gate IV:31) both describe the mechanism by which the righteous are fit to receive this reward — Tanya explaining that God grants the righteous strength through Torah so they are not nullified by the revealed divine light of the hereafter, and Nefesh HaChayim citing the Zohar's teaching that one who toils in Torah has thrones prepared for him in the World to Come.
Imrei Binah concludes that joy and wholehearted delight in the World to Come surpass the level of love and fear, because the mitzvot draw down the actual revelation of the divine essence, through which joy, awe, and self-nullification occur together.
Turning to the practical wording of the prayer itself, Revavos Ephraim (6:276) takes up the specific question of whether the phrase 'HaRachaman hu yanchilenu leyom shekulo tov' should be recited in Birkat HaMazon on Chol HaMoed, citing a letter from Rabbi Yosef Shmuelian that bears on this very question.