Star Trek Online explain it? Perhaps the development team can take and frame-by-frame it to have players mine, refine, and spend it. And that's just what's been done in a new development diary explaining to new players and returning players how dilithium works. It's the game's universal currency, in short, but it also serves to bridge the gap between having plenty of real money and having lots of in-game money with less real money.
Dilithium is rewarded to players through several means, but it's awarded as unrefined ore. The refining process is slow, with players having a soft limit of about 8000 refined dilithium a week (veteran and subscription players can refine slightly more). You can also trade it for the game's microtransaction currency on a live market. Once you have it, you can use it to buy new ships, rare crafting items, special gear, and certain Duty Officers. The full guide should answer any lingering questions you have about the currency, although it does not ask at any point if the players are down with dilithium.