Something or someone well established is well known, totally trusted and usually has a proven track record of success.
Becoming established doesn't happen overnight. You've got to put in the time and effort to prove yourself, make connections and build a good reputation. So it makes sense that this adjective is related to the verb establish, which means to "create, set up, or prove." Established is a handy word to know, especially when you're looking for a dentist — you probably won't want to try Dr. Lovedrill until his practice is more established.