Muun works by using a lightning service provider that pays LN invoices for users through a smart contract. This is known as a submarine swap. When you make a lightning payment on Muun, the bitcoin is sent on chain to the lightning service provider along with the LN invoice generated by the user. The LN provider has a time limit to pay the invoice after which they will receive the on chain funds once they can provide cryptographic proof they successfully paid the LN invoice. This way Muun only has to allow on chain transactions but through smart contracts with LN providers it can also make lightning payments.