Non-custodial wallets rely on the user to store their mnemonics. While often overseen, that step is essential to maintain your user funds long term since systems are programmed defensively to rather loose funds access then to loose user funds.
What are mnemonics
Mnemonics are a 12 word seed phrase that resolve your HD Privatekey. From that HD Privatekey, we can derive more than a billion private keys that each generate you a different address (for privacy purposes).
How do we keep track of paths
Since we have so many paths, how do we keep track of them? We do it with a peer to peer payment architecture, where each payment gets added to a UTXO array that involves the path descriptor. As long as we have the mnemonic, path and utxo we can always trigger a transaction (often from many paths and privatekeys at the same time).
How is that better than classical transactions
With the peer to peer model, we dont need to scan the whole blockchain for each wallet address (that can be tens of thousands that need to be checked against each transaction going through the chain). Instead we keep the data communication direct between sender and receiver (via paymail). Since we still support classical transactions for the foreseeable future, we are actually scanning the whole blockchain for transactions, but definitely will have trade offs in terms of speed and cost for our own infrastructure.