Where are they still in English? The state of your sensor.manfaser should be your translated text assuming:
sensor.moon_phase exists on your system;
The state of that sensor is exactly one of the translation keys you have provided — if it is not, your template will return the English version.
Where have you put that template YAML? Should be in configuration.yaml, and you should only have one template: key in the file. If this is your first YAML template, you will need to restart HA.
The template is in configuration.yaml and I have only one template.
As you can see from attached image I get English text … ‘Waxing gibous’ instead of ‘Tilltagande gibbousmåne’
sensor.moon_phase is not modified by your template, your template creates a new entity (sensor.manfaser?) which has the new value. So you would need to add that entity to entities card instead.