In iOS 17, one of the many features Apple added to the Messages app is an ability to automatically transcribe received audio messages, allowing you to read them instead when you aren't in a position to listen.
Another, smaller related improvement that Apple has made to Messages is to make it clearer where audio messages are kept when you choose to save them. In iOS 16, pressing and holding an audio message brought up a Save option in the pop-up menu, but after tapping it, it was not obvious where to go to locate the file.
iOS 17 fixes this issue by replacing the "Save" option with "Save to Voice Memos". After tapping it, all you need to do then is open the Voice Memos app, where you'll find the audio message at the top of the list.