The lines you refer to come as King Arthur addresses Sir Bedivere and instructs him about his final dying wish to see his famed sword, Excalibur, returned to the Lady of the Lake from whence it had emerged so long ago. Let us note what precisely Arthur says to Sir Bedivere in the lines you have specified:
For thou, the latest-left of all my knights,
In whom should meet the offices of all,
Thou wouldst betray me for the precious hilt...
Arthur therefore appeals to the value of loyalty and honour that should be shown by a knight to his liege in order to try and compel Sir Bedivere to fulfill his final commands and to return the sword. Arthur fears that Sir Bedivere will keep the sword for himself, and therefore needs to appeal to his sense of loyalty and honour in order to ensure that Sir Bedivere fulfills his final commands; commands that King Arthur is no longer well enough to carry out himself.