The joke is a pun. To say something 'humorlessly' is to say it in a serious way without any humor or jokes. 'Humorous' sounds like 'humerus' which is the name of a bone in the body, so 'humorlessly' sounds almost like they are 'humerus-less', as they are going to be missing a bone.
The doctors had to remove a bone from my arm , said Tom humorlessly .