For Apple TV 4th generation models and newer
While a video is playing, swipe down on the touchpad remote to bring up caption options at the top of the screen. Select the English option to enable captions. Swipe up on the touchpad to dismiss the caption options at the top of the screen.
For Apple TV 2nd and 3rd generation models
While a video is playing, hold down the select button (the silver round button at the top of the remote) for a few seconds and this will bring up a menu for captions. Turn captions to "English" (and not "Auto"). This will turn captions on when you return to the video. To return to the video, select the menu button on your remote to go back.