After uttering the first verse of Shema, it is an ancient custom to say "Barukh shem kevod malkhuto le'olam va'ed" – May the Name of the glory of His kingship be blessed for ever and ever.
But since this sentence is not written in the Torah, it is said in a whisper and not aloud, like the rest of the recitation (see Tractate Pesahim 56a).
A simple explanation for this is that after the grand recitation of Shema Israel, we add words of praise and thanks for being permitted, and able, to say this.
Viewed from a different angle, this sentence is the inner completion of Shema Yisrael.
The first verse of Shema speaks of God's unity in a way that negates the world's existence, for "there is none else beside Him."
In order to complement this, we add Barukh shem kevod malkhuto le'olam va'ed:
His glory fills all of reality, in place and time.
The very idiom shem kevod malkhuto – "the name of the glory of His kingship" – expresses a feeling of awe, for it is not God Himself that is mentioned here but rather His inspiration which fills the world, and even the mere "name of the glory of His kingship" is blessed for ever and ever.
It is as if we leave behind the perception of the sublime Unity to define God's kingship within the world.
And the reason why these words are uttered in a whisper is that we are not always sure that we are indeed worthy of being the bearers of "the glory of His kingship."
Only on Yom Kippur, when the Jews are like the ministering angels, do we say this sentence aloud.
—Rabbi Adin Steinsaltz
From an essay, “The Portion of "Shema" (Deuteronomy 6:4-8)” by Rabbi Adin Steinsaltz