When a Philips Hue bridge is connected to the internet it will periodically make a request to the Hue Cloud (provided by Signify) and register its local (internal) and external ip-address, together with the Bridge ID.
This data is exposed by the service available at discovery.meethue.com if, and only if, a request is made from the same public IP address as the bridge did when it made its request. If a request is made using another public IP address the response is empty.
The discovery service is used by Hue Log to automatically find out the local ip-address of your Hue Bridge. Hence it will only work if you run Hue Log on the same local network as your Hue Bridge is connected to.
Philips have implemented a very restricted rate limit for the discovery service. Only allowing a maximum of one request per 15 minutes. If this limit kicks in the discover service will respond with the error code (HTTP Status) 429 Too Many Requests.
If this happens the workaround is to manually register the bridge by finding the IP-address using the Hue mobile app, or simply wait a while until the discovery service works again.