The foundational obligation is rooted in Bamidbar 6:23, where the kohanim are commanded to bless the Jewish people, and the Rambam (Mishneh Torah, Prayer and the Priestly Blessing 14:11) derives from the word "koh" a set of conditions transmitted at Sinai — standing, raised hands, facing the people (panim k'neged panim), in the holy tongue, in a loud voice — establishing the face-to-face requirement as a foundational element of the rite.
The Shulchan Arukh codifies the practical sequence: the kohanim begin facing the ark and turn to face the people before reciting the blessing, and the Shulchan Arukh (Orach Chayim 128:10) records explicitly that they recite the brachah "asher kidshanu bi-kedushato shel Aharon v'tzivanu l'varekh et amo Yisrael ba-ahavah" — concluding with the word "love" — at the moment they turn to face the congregation, a formulation likewise recorded by the Tur (Orach Chayim 128).
The Magen Avraham (128:18) addresses the word "ba-ahavah" in the brachah directly, explaining that it is grounded in the Zohar's teaching that any kohen who does not feel love (rachamim) toward the people — or toward whom the people do not feel love — should not perform birkat kohanim, linking the language of the brachah's closing word to an interior condition of mutual love.
The Mishnah Berurah (128:37) makes this condition explicit and actionable: a kohen who harbors hatred toward the congregation, or who is hated by them, is in a state of danger if he ascends to the duchan, and therefore the brachah was formulated to conclude "ba-ahavah" — with love — precisely over this concern; if he cannot overcome the hatred in his heart, he should leave the synagogue before Retzeh.
Ein Aya (Berakhot 5:74) reflects on the spiritual logic underlying the whole structure: the raised hands (nesi'at kapayim) and the act of blessing together express that love of the community (ahavat ha-klal) fills the heart of the kohen dedicated to divine service, so that even his actions are consecrated to the welfare and flourishing of the people.
Peninei Halakhah (Likkutim I 11:3) adds that even a kohen who has sinned grievously is still commanded to ascend and perform the mitzvah, because it is ultimately God who blesses His people with love; the kohanim as a collective express the divine will that blessing flow to Israel, a framing that situates the "ba-ahavah" as rooted in divine love expressed through the kohen.
The Gemara in the Beit Yosef (Orach Chayim 128:18) cites the ruling of Rabbi Yitzchak from Sotah that the kohanim face the people while their backs are toward the Shekhinah, underscoring the orientation toward the people as a defining posture of the rite; and the Rambam (Mishneh Torah, Prayer and the Priestly Blessing 14:7) and the Magen Avraham (128:18) both affirm that the congregation in turn must direct their faces toward the kohanim — so the panim k'neged panim requirement is mutual, and Sotah 39a records Rabbi Elazar ben Shamua's testimony that he never raised his hands without a brachah, reflecting the gravity with which this act was observed.