ضَارِبٌ is an active participle on the فاعل pattern and means “one who strikes.” Under appropriate conditions, an active participle can govern like its verb.
In this sentence, أَبُوهُ is the one doing the striking, so it is the participle’s fāʿil. عَمْرًا receives the action, so it is the direct object. This is why the book’s definition includes both a verb and a descriptive form.