Display values of custom fields created on media entity in a template file I'm trying to display the values from the custom fields that have created on my Media Entity to my page--media.html.twig file. The fields that I'm trying to output are Location and Photographer

I tried using this code below, to make it work. 

{{ node.field_photographer.entity.value  }}

Also tried an alternative

{{ content.field_photographer.entity.value  }}
Your question is more about how to handle Drupal templates and passed variables. I suggest you to use Devel and Devel Kint modules to be able to debug variables just in your template like this:

{{ kint() }} - print all available variables

{{ kint(your_var_name) }} - print particular variable

And don't forget to enable twig debug mode - it will show you templates structure.

Once you have everything visible it becomes much easier to understand.