Slate is a type of gray metamorphic rock that is made up of quartz and other minerals. The rock can be split into thin layers and the slate used for things like roofing tiles.
In the past when paper was expensive, schoolchildren used slates to do their school work. These were usually made of actual slate rock. Students could practice their writing on the slate with a slate pencil. When they were done, the slate could be erased, which gave rise to the term "clean slate." If you've done things in the past you aren't proud of, you may hope to start over with a clean slate.