Why do we say Boruch Shem Kevod Malchuso L'Olam Voed in Kriyas Shema in a whisper?
- This was a declaration of loyalty by the Shevatim on Yaakov's deathbed, so it's a worthwhile prayer. But since it's not a verse found in the Torah, we say it quietly (Pesachim 56a)
- This was a prayer of the angels so it's not appropriate for sinful mortals to say it aloud, except for on Yom Kippur
From the Artscroll Yom Kippur Machzor, pp.69-70
In the Torah, the verse of Shema is followed immediately by the verse of Ve'ahavtah (Devarim 6:4-5). So in a way, BSKML"V might be a hefsek. The Tzlach says that the Gemara is telling us that by whispering it, we are not making a break between Shema and Ve'ahavtah. (see page 13 of this document for this and other explanations).
[The Tzlach does not address why we say BSKML"V out loud on Yom Kippur. He does discuss a custom certain people had to do so on Yom Kippur Katan. It seems that he originally defended the practice by saying that it was not an interruption in the words of Moshe (since it is after only one verse), but later on saw that the Maharsha explicitly said that it was an interruption.]
Another answer is given in the Midrash (Devarim Raba 2:25). The Midrash tells us that Moshe "stole" this prayer from the angels, and therefore we whisper it, to "hide it" from the angels. See here. This would explain why we say it aloud on Yom Kippur, since on Yom Kippur we are compared to angels, see here and here.
The Maharal in Nesiv HaAvoda says that baruch shem kevod is an expression of pure spirituality. Therefore to say it would be inappropriate because we are physical, having a body. To not say it would be painful to the spiritual side of us, the soul. Therefore, we whisper it.
Yaakov, who was talking for himself and was completely holy, was able to say it. Moshe, who was speaking on behalf of all of Klal Yisrael, who were not all on this level, did not say it.
On Yom Kippur when we shed our physical needs, even our body is uplifted to the point where it is appropriate to be expressed.
When reciting Krias Shema without a Minyan, Rav Chaim Solveitzik Zatzal of Brisk used to say "Baruch Shem" out loud, as he was of the opinion that the problem of saying it out loud was only when it is recited with a Minyan.
He obviously maintained the other reason, that the heavenly angels will be jealous of us saying such an exalted praise, and not the reason of the Mishna Berura 61:30 on Yaakov Avinu and the Shevatim. See Shu"t Teshuvos V'Hanhagos from Rav Moshe Sternbuch Shlita Vol. 2 Siman 46
In summary, when the Shevatim said Baruch Shem to affirm their belief in Hashem, they could confidently state that they were totally comitted to Hashem, as they had no Mitzvot other Emunah. We, after Mattan Torah, have plenty of Mitzvot, so stating out Emunah is not sufficient to bargain with Hashem for mercy. On Yom Kippur, we say it out loud as a means of saying, regardless of our lack of satisfactory observance of Mitzot, Hashem, please have mercy on us.