You can clean up a burnt cast iron skillet by sprinkling some coarse sea salt or Kosher salt into the cast iron skillet and then use a clean damp washcloth to scrub out the cast iron skillet.
You can also soak the cast iron skillet in some vinegar and water solution as well and add some of the above salt and then scrub it out wit the wash cloth.
Avoid using scouring pads or detergent scrubs as it can remove the seasoning from the cast iron skillet.