Action Details
Requirements and how-to articles for Discord's action:

Let your audience know that your server is official and gain access to extra features and perks (Discord is currently accepting applications for game developers/publishers, professional eSports teams, and Spotify verified music artists.)

You need this to get it done:

Basic Requirements
The basic requirements which every application must follow:
The applicant must... both the owner of the server and an official representative of the game/team/artist that is applying.
...include as much relevant information as possible to help us verify the validity of the application.
...use a formal email address related to the game/studio/team/artist to make it easier for us to understand that they hold an official position.
Additionally, the server should have channel and role structures as outlined in Discord's moderation guidelines.
Game Developer/Publisher Requirements
Server must be run by either the development team or the publisher behind the game in question.
The game needs to be live and playable (can even be just in open beta.)
Discord cannot and will not accept applications for servers based around mods for a pre-existing game.
eSports Organisation Requirements
Server must be owned by a representative of the eSports organisation itself.
Your team should have past experience being invited to a major offline tournament or LAN.
Music Artist/Label Requirements
Server must be owned by the music artist, or by a record label.
Server should be primarily focused on the music or the artist.
Currently only accepting Spotify verified artists with a healthy amount of monthly listeners.
Requirements missing or found errors? Report

How-To Articles
A list of useful links for you:

Discord Support - What Qualifies for Server Verification?
Official requirements for a server to be eligible for verification.
Discord Support - Verified Server Moderation Guidelines
Official moderation guidelines for verified servers.
Discord - Verification
Official information on server verification.
Link missing here or found errors? Improve How-Tos