Cleanse me with hyssop, and I will be clean;
wash me, and I will be whiter than snow.
Matthew Henry's Commentary explains the purging with hyssop this way,
"The expression here alludes to a ceremonial distinction, that of cleansing the leper, or those that were unclean by the touch of a body by sprinkling water, or blood, or both upon them with a bunch of hyssop, by which they were, at length, discharged from the restraints they were laid under by their pollution."
You see, sin restrains us from fellowship with God. We cannot have access to him because we are unclean, but he will purge us and cleanse us, so that the access can be restored.
The clearest understanding of this is the cross. Only by the blood of Christ are we truly cleansed and as Isaiah put it, "...made whiter than snow."