Why I don't need to COMMIT in database trigger?

You get an autonomous transaction context for these triggers automatically.

From the CREATE TRIGGER docs:

One or more particular states of the database that can cause the trigger to fire. You can create triggers for these events on DATABASE or SCHEMA unless otherwise noted. For each of these triggering events, the database opens an autonomous transaction scope, fires the trigger, and commits any separate transaction (regardless of any existing user transaction).

What happens if the trigger fails depends on the exact trigger/event. See Exception Handling in Triggers. In particular, a logon trigger that fails with an exception might very well lock non-dba users out - the exception makes the login fail unless the user has specific privileges.