The fact that you say it is connected to your router (I assume via WiFi) it sounds like you have the CAD (Consumer Access Device) enabled version (meaning that the display does actually make the data available) which is the good news.
What is not clear from doc is how it makes that data available (the bad news). The doc mentions a TLS secured MQTT connection with JSON based messages. All that sounds great, except it doesn't say if the device is a MQTT broker or a MQTT client. If it is a client then the question is what broker does it publish the data to and is it configurable. If it's a broker, do you need to know specific details (username/password/client certificate) to connect.
My interpretation of the doc is that the device is a MQTT client and that it publishes to a pre-configured MQTT broker in the cloud based on the following:
The IHD6-CAD-PPMID sends the user’s real time smart meter data securely to Chameleon’s cloud platform via the Chameleon CAD API
If this is the case then you probably need to talk to your energy supplier for details of how to access this API to retrieve your data.