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The Xbox One and the Xbox 360 both support subtitles, as long as the title itself has subtitles. The process is identical for both systems. The "Audio & Subtitles" option will appear. You'll now be able to select your desired subtitles. The subtitles will appear as soon as you select them. If closed captioning is enabled system-wide, subtitles will appear in Netflix even if they are disabled for that title. Xbox 360 - Press the Guide button on your controller and open the "settings" menu. Select "System" and then "Console Settings." Select "Display" and then the "Closed Captioning" option. Select "Off" to disable closed captioning system-wide. You should be able to return to your video and watch it without subtitles. Xbox One - Press the Guide button on your controller and open the "settings" menu. Select the "Closed Captioning" option and then select "Off." Your video in Netflix should no longer have captioning.

Summary:
Start playing the title you want to add subtitles to. Press Down on your controller while the video is playing. Select "Audio & Subtitles" and press A. Choose your subtitle options. Disable closed captioning on your system if subtitles won't turn off.