How are better exposed filters - checkboxes value - assigned
This has nothing to do with Exposed filter settings, it's to do with what information is stored in the Taxonomy term Entity Reference field on your Content Type, and it's the term ID, not the term name.
So when you add the most ready-available Taxonomy Fields/Filters to your Content Views you will get the term ID.
Other Taxonomy term information (term name included) is stored in another database table and you have to add the Relationship for that particular field in order to pull information from that table into Views.
What you want is to add the Relationship for the Field on your content that stores the Taxonomy term reference. You can require the Relationship if you only want to see nodes in results where that field isn't empty.
After you add that Relationship you will get more options under Fields and Filters, one of those options is the Taxonomy term Name. They will use the Relationship by default since they can't be added to the Views without it.
Add the Name Filter to your Views and expose it. Adjust the Vocabulary settings and other options to your particular need.