What is the special database "postgres" for?
There is nothing special about it. As described in PostgreSQL documentation, Creating a Database, you need to be connected to a database, in order to create another one:

Since you need to be connected to the database server in order to execute the CREATE DATABASE command, the question remains how the first database at any given site can be created. The first database is always created by the initdb command when the data storage area is initialized. (See Section 17.2.) This database is called postgres. So to create the first &quot;ordinary&quot; database you can connect to postgres.

and in Template Databases:

Note: template1 and template0 do not have any special status beyond the fact that the name template1 is the default source database name for CREATE DATABASE. For example, one could drop template1 and recreate it from template0 without any ill effects. This course of action might be advisable if one has carelessly added a bunch of junk in template1. (To delete template1, it must have pg_database.datistemplate = false.)
The postgres database is also created when a database cluster is initialized. This database is meant as a default database for users and applications to connect to. It is simply a copy of template1 and can be dropped and recreated if necessary.