The Torah tells us that on Rosh Chodesh Nissan, the eighth day of inaugurating the Mishkan, Aharon Hakohen was charged with the offering of the special sacrifices[1]. At the conclusion of Aharon's offering of these korbonos tzibbur[2] Aharon recited the priestly blessings (Vayikra 9:22). The Talmud (Sotah 38a) points out that based on this statement it would appear that the halacha is that duchenin (the mitzvah of reciting birchas kohanim - the priestly blessings) can only be fulfilled at the conclusion of the offering of the korbonos tzibbur.
This Talmudic statement led Rav Yaakov Emden to comment that the fact that we still practice this mitzvah today, even though we have not offered any korbonos, is only miderabonon; Min haTorah, the mitzvah of duchenin must be connected with korbonos. The Mishna Brura (vol. 2 pg. 19 in footnote) disagrees vehemently with Rav Yaakov Emden, and proves from many sources that even today the mitzvah to duchen is still min haTorah[3]. The Mishna Brura does not, however, deal with the possuk cited above.
The solution to this problem was given by two brothers, who were both prominent geonim in the nineteenth century, in their respective seforim: Rav Yaakov Karliner in his Teshuvos Mishkenos Yaakov (Orach Chaim, siman 66) and Rav Yitzchok Bruchin in his sefer, Keren Orah (Sotah). The Talmudic statement (Berachos 26b) that Tefilos (prayers) are considered as if we had offered korbonos is a biblical principal. When an individual davens, it is considered as if he had offered a korban yachid. When the chazzan recites the tefillah out loud representing the entire tzibbur, it is considered as if a korban tzibbur had been brought. And indeed, it is at the conclusion of the chazan's tefillah that the kohanim fulfill their mitzvah to duchen. Their obligation to duchen at that point is min haTorah, since the conclusion of the chazzan's prayers is biblically equivalent to the conclusion of the offerings of the korbonos tzibbur!
When we duchen on yomim tovim, we all recite the piyut "v'se'erav": that Hashem should accept our prayers and consider it as if we had offered actual sacrifices. This piyut is inserted in the middle of the chazzan's recitation of the beracha of retzei, which is referred to in the Talmud (Megillah 18a) as "avodah", the literal translation of which is "the offering of sacrifices". The piyut just makes more explicit the simple meaning of that beracha, that our prayers should be considered as if we had actually offered sacrifices. Because it is the theme of this particular beracha that makes it possible to fulfill the mitzvah of duchenin, the rabbis required (Sotah 38b) that the kohanim must at least begin to go towards the duchen (the platform where they will recite the birchas kohanim) by the end of this beracha.
Rav Soloveitchik added on (in a yahrzeit shiur) that the beracha of retzei is not simply a repeat of the immediately preceding beracha of shema koleinu. In shema koleinu we ask Hashem to accept our tefillos. In retzei we add a request that Hashem should accept our prayers as if they were a sacrifice. The term "ritzui" is a technical halachik term appearing most often in Tanach in connection with acceptance of sacrifices. In mishnaic Hebrew as well "hurtzah" means the sacrifice is "kosher" and is accepted; as opposed to "lo hurtzah" which means that the sacrifice is "not kosher", i.e. not acceptable (see Beikvei Hatzon, pg. 82). The gemara (Bearchos 22b) disqualifies one's prayer in a specific instance, and requires that he daven all over, based on the principle of "zevach reshaim to'eivai - the sacrifice of the wicked is an abomination" (Mishlei 21:27). The equation between the tefillah and the offering of the korban is taken very seriously; it was not intended as a mere metaphor[4].
[1] As opposed to the first seven days, when the special sacrifices were offered by Moshe Rabbeinu
[2] Communal sacrifices, as opposed to korbonos yachid, which are offered on behalf of an individual person
[3] See also Binyan Shlomo (teshuvos of Rav Shomo Vilner) siman 10
[4] In Shulchan Aruch (Orach Chaim, end of siman 76) the same halacha appears regarding kriyas shema: we sometimes disqualify one's recital of shema (due to the principle of zevach reshaim to'eivai), and require that shema be repeated. The students who attended the yahrtzeit shiur found it difficult to understand why this principle should be extended even to krias shema, according to the Rav Soloveitchik's understanding of the gemara.