Discord is now accessible on Xbox platforms, having previously only been available on PC and mobile. The voice mail function, exclusive to Insiders since July, is now accessible to all gamers.
Download the Discord mobile app, then connect your Xbox account to it (Account > Connections > Add) to complete this task. Of course, you will need to do it once again if you have completed it. All that remains to be done when this is finished is to join a call or voice channel and transfer it to your console. This link also enables the game on Xbox to be shown on Discord.