Also known as the "Egg Room", this multi-chambered construct is actually a place where people can go to converse privately, either through voice or through the KI's text messaging system without being overheard by those outside the doors of each chamber. Each door operates on a timer, and will open again after 5 gorahntee, or 3 minutes.
The large floating "egg" in the center of the Private Room complex is, in fact, a seed; possibly depicting the seed of the Great Tree of Possibilities, or the Great Tree of D'ni. It is possible that, in addition to ensuring privacy of communication, the chambers were also used (or intended to be used) for meditation and prayer.
It should be noted that in Uru: Complete Chronicles, the Bevin Private Room is used as an easter egg, and should not be considered an accurate depiction of the room as it exists in Myst Online: Uru Live.