How do I save audio messages on my iPhone?

The iOS 8 feature that allows you to send audio texts via the iMessage app is set to automatically delete any voice texts 2 minutes after the recipient listens to the message. This is necessary to save space on your device, but you can choose to keep your audio messages and delete them yourself. To stop your audio texts from expiring navigate to Settings > Messages and tap the Expire panel in the Audio Messages section. From here you can choose to have your audio messages disappear after "2 Minutes" or select the "Never" option so they remain on your device until you manually delete them.

iOS 8 Audio Messages Settings

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In general, iTunes supports to backup your iPhone audio messages to computer. However, iTunes does not allow you to access and extract data in iTunes. If you want to extract iPhone audio messages directly from iPhone or iTunes, you could use some third-party software, like iRefone to help you.