Twitch is an app on Xbox One. Boot up your Xbox and search for Twitch in video apps. Set up and launch the app and you should be good to go broadcasting on Twitch, no laptop or extra equipment required.
If you did want your gameplay on your laptop, you can buy a capture card from a company called Elgato. There is also a company called Hauppage that sells a cheaper one. They plug into the monitor and a USB port.