A quick google for a C++ HTTPS/SSL Client delivers this: http://www.example-code.com/vcpp/ssl_client.asp
You send the request, you get the reply. The reply, as stated, is in JSON. A quick google for JSON parser in C++ leads me to: https://github.com/miloyip/rapidjson
Add in a dash of: https://github.com/justintv/Twitch-API and you should be on your way.