Open your phone’s voicemail app, then tap (or in some cases, tap and hold) the message you want to save. You should be presented with a list of options; the save option will usually be listed as “save”, “save to phone,” “archive,” or something similar.
On the iPhone, go into voicemail and tap "Greeting". Record a few seconds of something and tap stop. Before you tap save, use iFuntastic or SFTP to navigate to /var/root/Library/Voicemail on your iPhone. You'll see a file called "Greeting. amr". This is the audio you just recorded. Delete that file and replace it with your custom message which should then be renamed to "Greeting.amr". Tap "Play" on the iPhone to verify your message works. If it plays, tap "Save" and your custom message will be uploaded to AT&T's servers.

Alright let me try to clear this problem as much as I can. From the parent carrier, we have a limitation or you can say a requirement that all numbers should have the Voicemail set up. So what we do for those customers who don’t want it, we disable it from our side but again the numbers keep syncing from the parent carrier’s network from time to time (roughly once a quarter). When the numbers get synced, voicemail automatically gets enabled. To some people, this is alright but sometimes it becomes inconvenience for the customers to call us again to disable it.
With Visual Voicemail on iOS 9 you can listen to a Voicemail easily and you can also save or send a message on to someone else through iMessage or email.