Here, the reader is introduced to the idea that death is not something which happens all at once, but is rather a gradual process, one which must be "adjusted" to. This adjustment is a parallel of the rest of the world adjusting to her death. Relatives still bringing flowers probably, still talking about her, still going through her things.
The next section goes into the fact that high ideals of life are of little use once one is dead. Here one who lived and died for truth finds himself in the same place as one who died for beauty. The first thing the one in the adjoining room did was "questioned softly ‘why I failed’". The only way they identify themselves is by the way they died. Likewise, that is all they are remembered by on earth. To a living person, it does not matter what they died for, the mere fact that they have died is all that is what is important.
The last four lines emphasize that all dead are the same to those living. They were "as Kinsmen, met a Night", grouped into the same category; dead. They are remembered for a while, represented as they "talked between the rooms", but eventually are forgotten by the world. First people stop talking about them altogether, and forget their lives "the Moss had reached our lips" and then even the names of the dead are forgotten "and covered up – our names –