It means "like a diplomat": that is being polite and making every effort to avoid giving offence. "'Did you enjoy the meal?' ... 'Yes, I found it very unusual' was her diplomatic reply.'" It can also be used more literally as in "The crisis in [Ukraine] has lead to much diplomatic activity".

"Diplomat" comes from the same root as "diploma." A diplomat is a person, authorized by a formal written paper ("diploma"), to speak on behalf of his government. Diplomats learn to speak very carefully to avoid giving offense, because if someone gets angry at them it could start a war. Being "diplomatic" means "phrasing your words to sound nice and avoid hurting peoples' feelings." It means "to be tactful." Suppose someone said to me "Tell him no, but be diplomatic about it." I might say to him. "I very much regret that unfortunate circumstances beyond our control make it difficult for us to grant this specific request at this particular time, but rest assured we always are glad to help whenever we can, so please stay in touch and perhaps at some future there will be some other matter we can assist you with." He thinks of a rude and angry reply, but, being diplomatic, says "I am disappointed that you can't see your way clear to help me, but as always it is a great pleasure to see you and please give my regards to your lovely wife blah blah blah blah..."