Treasure is the multisignature analog of a wallet. Unlike a regular wallet that has only the private keys and public keys of one person, a treasure has the public keys, but not the private keys, of N different people, M of which are required to sign transactions. It is still necessary for members of a treasure to store their private keys in a wallet.
The purpose of treasure is to make multisignature transactions as easy as regular transactions, or nearly so. Spending bitcoins from a treasure should be as easy as spending bitcoins from a regular wallet. The only extra hurdle is that other people are required to sign the transaction before it is broadcast. This means there is an extra stage.
Normally, you sign a transaction and broadcast it.
Sign -> Broadcast
But with treasure, rather than simply spend bitcoins, what you do is propose that some bitcoins be spent. Then you can see the transactions that you or others have proposed, and optionally sign them. Once a transaction is fully signed, it is broadcast.
Propose -> Sign -> Broadcast