Staining on surface of slate
The residue is Quarry dust & cutting residue & is easily cleaned off with just plain water.
You may need to do this a couple of times, making sure you dry off in between using a soft, lint-free cloth.
When you are happy with the finish, you can then apply your sealant.
It is important that all residue is thoroughly cleaned off & the slate is completely dry before you seal, or you will 'trap' any marks into the sealant.