Voicemail to email is a system feature that converts voicemails to audio files and sends them to your inbox as an email attachments, where you can easily play and listen to them. This gives you both access to your voicemails from anywhere you have online access as well as records of basic caller ID info - duration of the voicemail and the number it came from.
The feature is not enabled by default. It must be set up in advance.
When you listen to a voicemail on your phone you get the option to delete it. But when you open and play the voicemail attachment it will still remain on your phone. This may require you to listen to voicemails again from your phone before you delete them.
To address this, every voicemail email also comes with a link that allows you to delete it from your phone. Simply click it once you listened to the voicemail to keep your mailbox tidy.