These are special cryptographic entities provided by Olympus server in exchange for a Lightning payment. You can think of them as one-off tickets which can be used to anonymously store such data as channel backups and scheduled Bitcoin transactions on Olympus server.
Currently an automatic Lightning payment of 2000 Satoshi to get 50 storage tokens is issued by wallet when a first payment channel is created. Once obtained these tokens are stored in a wallet while being spent gradually and automatically as the need comes by.
There is no incentive for me the developer to maintain a public Olympus server without getting paid for it somehow. Accepting micropayments for some useful API calls seems like a good idea but using a Lightning payment per each call is an overkill.
The goal is to keep Lightning Wallet as private as possible so any kind of traditional approach like registration and user accounts is a no-go. But at the same time 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 and technically speaking these storage tokens are blind signatures which posess an unlinkability property so every API call made by your wallet is anonymous and private.
All that said, you can opt out of this at any moment by setting up your own Olympus server which would use a free public key based authentication instead of paid tokens.