Baruch shemo or baruch hu shemo in this context means “Blessed is God’s name.” (Literally, it’s “blessed is his name” but of course God has no gender.) It’s a little addition that some people like to make to the blessing, when the blessing includes the Name of God, or rather, the stand-in for the Name.
The Hebrew name of God, the Tetragrammaton [τετραγράμματον is Greek for “four letter word”] is never pronounced aloud. It is spelled yud-heh-vav-heh, but we no longer have the vowels to pronounce it. Moreover, tradition has forbidden we say the Name since at least the time of the Mishnah (c. 200 CE,) and probably long before that. Instead, observant Jews make substitutes for the Name, and sometimes substitutes for the substitutes:
Instead of the name, in prayer we use the word Adonai (“my Lord” in Hebrew.) Some observant Jews do not use even that name aloud except in prayer, and in normal speech substitute Hashem (“the Name” in Hebrew.) Reform Jews commonly use “Adonai” but still avoid pronunciations of the Name itself.
But what about Baruch shemo? It’s a further way of paying respect to the Name of God. When in a blessing the shaliach tzibbur (service leader) says “Adonai” (the stand-in for the Name, remember?) some individuals may say “Baruch shemo“:
Service Leader: Blessed are You, Adonai —
Congregant: Blessed is God’s Name!
Service Leader continues: Our God, Ruler of the Universe…
In congregations where this response to the Name is common, service leaders often pause slightly for it, so that it will not obscure the rest of the words of the prayer.
Jewish prayer is active and interactive. We sing, we chant, we have choreography, and depending on the custom (minhag) of the congregation, there is room for improvisation. This is one example of the way that Jews make the traditional prayers our own.