This quote is an implicit Biblical allusion to Pontius Pilate's speech when he delivered Jesus over to the crowd for public scourging, then crucifixion. Pilate abnegated his responsibility by doing so, since it was within his jurisdiction (and not that of the Jewish Sahedran Court)to decide Jesus' fate. By washing his hands in public, Pilate absolved himself from any guilt since the prisoner would be executed for religious heresy rather than political treason. It was his way of "passing the buck."
As Pilate, Lady Macbeth hopes that a symbolic act of purification will cleanse her of her responsibility. She has confused metaphor with matter! If she later feels remorse, it is in a relative kind of way. One gets the impression that she is more focused on the guilt she must bear rather than upon the actual suffering she has inflicted. She loses her initial brazeness:
In contrast to Lady Macbeth's forceful disposition on the first three acts of the play, her actions in the last two acts are much less confident or ambitious. Lady Macbeth in the sleepwalking scene appears to be tormented by her knowledge of Macbeth's actions. In V.i, Lady Macbeth reviews the various crimes her husband has committed and appears to be attempting to wash blood from her hands. This scene contains Lady Macbeth's famous "Out damn'd spot!" (V.i.35) speech.
-from enotes, 'Lady Macbeth, Character Analysis'
Here, once again, there is a disassociation between the crime and the criminal. Lady Macbeth would "do laundry," so to speak, but refuses to see that the true "stain" is within herself. Her statement is ironical in that the handwashing scene forebodes the 'bloodbath' which follows.