I believe (admittedly from skimming and educated guesses) the setup is that the League of Entropy network is a set of computers operated by different people which each have published public keys, and the participants regularly do multi-party computation to create random numbers that provably can not be manipulated by any non-majority subset of the group. Besides calculating verifiably random numbers, the computers will also decrypt content for anyone if you send it content encrypted to their public keys and the encrypted content contains a time range that includes the present. (Maybe their main protocol for random numbers requires this protocol to be implemented?) Then you can do time-lock encryption by encrypting some content, splitting the key using a threshold scheme, then separately encrypt each part of the key to a different participant of the League of Entropy network bundled with a requested time-range for decryption to be allowed. It works as long as enough League of Entropy network participants stay online and enough of them don't defect or get hacked.