Add a security deposit to your listing, first and foremost. It discourages people from causing damage to your listing when they know they've already had a hold placed on their card.
Speak to them about the damage, but do so within the AirBNB messenger. This way, all conversations are in writing and can be reviewed by AirBNB staff. Say that after they checked out,. you observed X damage to X objects, and include photos in your message. Give them an opportunity to respond - they may say yes, we accidentally broke it, etc. In this case, you'd have their admission in writing and could go ahead and make a monetary request immediately. AirBNB would reimburse you for the damages - have a justification prepared for the amount you're asking, if you do not have any original receipts. If they deny it, you can still make the monetary request but it might be a little trickier to collect. If they refused to pay, AirBNB would get involved to mediate the dispute and would likely settle somewhere in the middle.
