I haven't studied this in depth, but as far as I can tell it works as follows.
Mixin Safe is a 2-of-3 multi-sig.
There is one key held by you, the owner key.
The second key is held by your family/friends/colleagues/other trusted contacts, called the members key.
The third key is held by Mixin Safe themselves, but is timelocked for a year, called the recovery key.
You can spend coins using your key and the key held by your trusted contacts with their approval. If you lose your key, or your trusted contacts lose their key, you can recover your coins after one year with the key you do still have and the recovery key.
I would also say that I will never use such a product, for a couple of reasons. Personally, I do not want a third party involved in my storage, and I certainly don't want to be paying a third party to be involved in my storage. I also highly value my privacy, and don't want a third party being able to see all my holdings and transactions. I know there is a market for such products given the recent Ledger Recovery nonsense, but that market is not me.
However, on poking about the website a bit more I have one main concern, and it revolves around the members key. How does it work exactly? It is a multi-sig embedded in a multi-sig? Is it SSS? How do the threshold number of members come together in order to recreate their key? Can I pick the threshold? Your pricing model says you charge $20 per transaction(!). How can you enforce this when I am supposed to be able to access my key and the members key without you? If the members truly did hold this key, then I can recover my multi-sig to any wallet and make transactions without paying your fee, no? Something doesn't add up.
It's the CMP-MPC protocol from Fireblocks, so there is no private key for the members key, and it has never existed. But n members hold some shares, they can sign the message with their share and combine the signature to form the final signature.
Technically and actually you can do all of this for free. What we charge is that we make a tool for you to easily do all these multisig and MPC stuffs easily. We sell wallet software, just like people sell wallet hardware, you can manage your private key without the hardware.
Mixin Safe is completely different from Ledger Recovery, they are trying to backup your private key, Mixin Safe never gets access to your private key. What we promote is multisig+timelock, the ultimate technology in Bitcoin, to help people preserve their coins without single point of failure.