Monster Stones are used to pay for Automaton repairs at Dampé Studio. You'll find and pick up many of these purple stones during your adventures in Hyrule.
If your Automaton breaks, then you can warp to Dampé Studio and use Monster Stones instead of rupees to pay for the repairs.
If you want to know more about Automatons, check out our complete Automaton Guide, along with how to unlock every single Automaton in Echoes of Wisdom.