But what exactly is that?
Firebase Cloud Messaging (FCM) is the latest version of Google Cloud Messaging (GCM), which is Google's Push Notification Service.
It allows developers to send data from their App Server towards the Client App. For your case, you may be able to send over the desired data towards your Client App, by integrating your Database with FCM.
How does is interact with the database? How do I use it for listening and retrieving data?
If you are using Firebase Realtime Database, you may choose to integrate with FCM in such a way (referring to @FrankvanPuffelen's answer):
Sending messages to devices based on inserts into the Firebase Database will require you to run a trusted process, typically on an app server that you control. This trusted process listens to the database changes and then calls Firebase Cloud Messaging to send the messages.
For an example of how to send messaging from a node.js script, see my answer here: How to send Firebase Cloud Messaging from a node server?
In summary, you don't use it to listen for data. You can however use it as the medium to send the data from your Database towards your Client App.