The Enclave is a way for users to lock their funds in a vault, and have privacy pertaining to their ownership and transfer activity. When in the vault, funds can be privately transfered. The funds can then be withdrawn from the vault. Your funds remain in your control and on your host blockchain.
How does this work?
Using the Sapphire Trusted Execution Environment, a user has their deposit transaction bridged using the Celer Inter-chain Message Framework. This is the only part of the process that requires an external bridge, the rest is entirely trustless. Once the funds are locked in the Enclave, the user needs to set a separate public key for their encrypted reads. This allows for information to be privately shared with the user, such as their current account balance.
They can then use their current wallet to "sign" transactions. This allows their approval of a transaction to be trustlessly relayed and encrypted before being submitted to the secure enclave. This relaying is entirely trustless. They can also submit the transactions to sapphire directly if they prefer. This decreases visibility because the transactions can be relayed by any party. So a transfer from wallet A to wallet B will be private, will not come from any specific origin, and can not be identified as being a transfer event.
This creates an end to end encrypted and private Enclave on Arbitrum or Ethereum for standard ERC20 transactions.