"Lesen" is the verb "to read." Literally translated the phrase yeilds: "on again reading."
It has been suggested that "Auf Wiederschreiben," using the verb "to write," would be more appropriate. The "-lesen" ending is considered more aesthetically pleasing when compared with sister phrases Auf Wiedersehen and Auf Wiederhoren.
This is a recently coined phrase for use in the online text environments of IRC and e-mail.