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 and scheduled Bitcoin transactions on Olympus server.
A Lightning payment of 2000 Satoshi 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.
Storage tokens help to pay for the maintenance of a public Olympus server. Accepting micropayments for some useful API calls seems like a good idea but having a Lightning payment for each invidividual API call is overkill.
Lightning Wallet aims to be as private as possible, so no registration or user accounts are required. However, simply allowing anonymous users to store any amount of data on a server is an obvious attack vector.
Storage tokens solve both of these issues beautifully: a single Lightning micropayment gives you an access to 50 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.