Two ways:
1- You go to their "Mining" tab and just click "Start mining". That's it. Mining happens in your browser in the backend through hashing. You can even increase the hash rate in settings
2- If you're a website owner, you implement it; so when visitors browse your site, you earn JSEcoin.

You can also earn by referring others. Honestly, I like the idea of JSEcoin quite a lot and I think that's what's going to matter in the long run.