When you need to rely on Windows Server Backup, you want some sort of reporting. A lot of scripts with mailing functions are available (e.g. WSB Backup to network with email notification and rotation) . If you want to rely on simple messaging from the server itself, you can use scheduled tasks for this.
Windows Server Backup writes events to its own event log (event logs > Applications and Services Logs > Microsoft > Windows > Backup > Operational). We can use these successful (4) and unsuccessful (5) events as a trigger.
Create two scheduled tasks (can run under system) with the following triggers:
In the action of the tasks, run a powershell command to send a mail.