Quotation על במדבר 15:35

Blessed [is His] Name, Whose glorious kingdom is forever and ever.²Jacob wished to reveal to his sons the ‘end of the days,׳ whereupon the Shechinah departed from him. Said he, “Perhaps, Heaven forbid! there is one unfit among my children.” His sons exclaimed to him, “Hear Israel” etc., just as there is only One in your heart so is there only One in our heart. Ben Lakish explained that Jacob said, “Blessed [is His] Name” in response to his sons’ exclamation. The Sages wondered, “Shall we recite it aloud when our teacher Moses did not say it? Shall we not say it at all, when Jacob ordained it?” Thus, the Rabbis enacted that it be said quietly.—Maseches Pesachim 56a.
When Moses went up to the heavens, he heard the angels praise God by saying, “Blessed [is His] Name, etc.” When he returned to earth, he taught it to the Jewish People. We say it in an undertone so as not to offend the angels. On Yom Kippur however, when we are considered like angels, we are permitted to say ברוך שם aloud.—Midrash Rabba to Deuteronomy