These are special cryptographic entities provided by Olympus server in exchange for a Lightning payment. You can think of them as single-use tickets which are used to anonymously store data such as lightning channel backups or watch revoked channel states on Olympus server.
A single Lightning payment is automatically made by the wallet in exchange for 50 storage tokens when your first payment channel is created. Once obtained these tokens are stored by the wallet and are spent gradually and automatically when needed.
Lightning Wallet aims to be as private as possible, so no user account is needed to store your data on Olympus server or use a watchtower.
However, simply allowing anonymous users to store any amount of data is an obvious attack vector. Some kind of pay-per-use scheme is required but directly using a Lightning payment per each storage API call would be an overkill.
Storage tokens solve exactly this issue: a single Lightning payment buys 50 tokens and thus gives you an access to 50 storage API calls. Technically speaking, storage tokens are blind signatures which posess an unlinkability property, meaning every API call made by your wallet is anonymous and private.