The BT (Ber. 55a) cites R. Yitzhak who stated that one should not be appointed over a community without having the community consulted with first. For precedent, he refers to a communication between God, Moses and Israel on the appointment of Bezalel:
אמר רבי יצחק אין מעמידין פרנס על הצבור אלא אם כן נמלכים בצבור שנא' (שמות לה, ל) ראו קרא ה' בשם בצלאל אמר לו הקדוש ברוך הוא למשה משה הגון עליך בצלאל אמר לו רבונו של עולם אם לפניך הגון לפני לא כל שכן אמר לו אף על פי כן לך אמור להם הלך ואמר להם לישראל הגון עליכם בצלאל אמרו לו אם לפני הקדוש ברוך הוא ולפניך הוא הגון לפנינו לא כל שכן
Trans. (from Sefaria):
Rabbi Yitzḥak said: One may only appoint a leader over a community if he consults with the community and they agree to the appointment, as it is stated: “And Moses said unto the children of Israel: See, the Lord has called by name Bezalel, son of Uri, son of Hur, of the tribe of Judah” (Exodus 35:30). The Lord said to Moses: Moses, is Bezalel a suitable appointment in your eyes? Moses said to Him: Master of the universe, if he is a suitable appointment in Your eyes, then all the more so in my eyes. The Holy One, Blessed be He, said to him: Nevertheless, go and tell Israel and ask their opinion. Moses went and said to Israel: Is Bezalel suitable in your eyes? They said to him: If he is suitable in the eyes of the Holy One, Blessed be He, and in your eyes, all the more so he is suitable in our eyes.
Based on this, R. Hai Gaon (cited in Shibulei Haleket §140) explains that when one is appointed to lead the birkat ha'mazon (grace after meal), typically a kohen or distinguished individual, the group charges him by saying "bless", as if to say 'you are worthy before God to lead [because of your stature], surely we must too find you worthy'. The leader then begins "with your permission let's bless..." insinuating, 'you must agree [to my leading] because God is in agreement with you'. Whereupon the group responds "with permission of Heaven", as if to concur, 'we are in agreement [to your appointment] so that Heaven (God) should accede that you are befitting before us'.
Although R. Zedekiah in his Shibulei Haleket cites the above regarding the appointment for breaking bread, it is no less the origin and reasoning for its recitation for birkat ha'mazon too. Indeed, R. Shem-tob Gaguine (Keter Shem Tob, vol. 1 142f.) referred to the above in his section on birkat ha'mazon and offers it as explanation for there as well.