You install an app on your Android or IOS device, then add the Blynk library and a few lines of code to your ESP8266 sketch. No need for a public IP address, just a connection to your local wifi.
Blynk also allows you to send commands from your device to the ESP, so you could toggle an LED or turn on a fan, for example. It's not as robust as thingspeak, but it's pretty cool for what it does, it's free, and it won't take you more than 30 minutes to get a demo running.