I believe it means 'to have a fresh start with a clear conscience'
A person has committed a crime but they have made up for what they have done wrong and intend to now live an honest life then they could be described as starting over with a clean slate.
Slate was used to write on many years ago so it refers to the idea of having a clean conscience to move forward without the past dragging you down.
It may refer to other situations but this is one that comes to mind.
PS Another version of 'clean slate' is the Latin tabula rasa - which is still sometimes used in Br. Eng, and refers to an earlier writing technology. The tabula was a sort of tray filled with wax. The writer scratched the characters with a stylus, and shaved of the top layer (think of 'razor') in order to start afresh.