After we say the first pasuq "HaShem is our God, HaShem is one", we read a sentence which is not part of the Biblical text of the Shema Israel: "barukh shem kebod malkhuto le'olam va'ed". "May the Glorious Name of His Kingship be blessed forever". We dare to utter God's Name, in the same way we pronounce any name. That is why we disclaim by this phrase that the understanding of His Name lies beyond our human capacity. This phrase is considered a very high form of praise, where we praise God, by praising the glory (=the ineffability) of God's Name.
The rabbis in the Talmud tells us that this phrase was first said by Yaacob Abinu: In his deathbed he gathered all his children to find out if all of them were following his footsteps. When they said: "HaShem is our God, HaShem is one", he replied: "barukh shem kebod..." as an expression of gratitude, peace of mind and praise to HaShem knowing that all his children were following his ways.
In the times of the Bet-haMiqdash, this phrase was said by the whole community of Israel gathered there for Yom Kippur, when the Cohen Gadol (High Priest) pronounced the ineffable name of God of four letters. Besides being one of the highest expressions of praising to His name, barukh shem kebod... also served to hide the Name of God, by saying it out-loud by all, while the Name of God was pronounced by the Cohen Gadol.
Every day when we say the Shema Israel we read barukh shem kebod silently, whispering it. To signal that it does not belong to the original Biblical text. Except for one day in the year. On Yom Kippur we say barukh shem kebod malkhuto out-loud in remembrance of what took place in the Bet-haMiqdash.