There is a chance you find your answer in the 'supposedly duplicate' question someone has linked to in the comments.
Failing this, the only way that I have found an answer to this is by actually visiting the airports website. Most major ones will have a section that displays a list of all the airlines that fly to-from that airport. Most of these lists will also contain a list of destinations directly reachable using these airlines, unless the list is super-huge!
See for example Gatwick Airport: http://www.gatwickairport.com/flights/destinations/