Facebook gives you the option to save audio messages to Facebook Messenger. It used to be possible to do this from a phone or tablet, but now you can't save your messages only from a computer. To download your messages, you will need to open Facebook in mobile mode with your computer's browser and open your audio messages. There you will find a menu that contains a download option.
Steps

Step 1. Open
The only way to download an audio message from Facebook is to download it from the mobile version of their website. It might sound weird, but as of April 21, 2021, all other download methods have been removed.
It won't work with your phone's browser, you have to be on a computer

Step 2. Log into your Facebook account
If you are not already signed in, follow the on-screen instructions.

Step 3. Click on the Messenger icon
It's a speech bubble icon, located at the top right of the search bar. This will open the list of your messages.
For some people, the icon may be at the top and in the center

Step 4. Open the conversation
Click on the conversation that contains the message you want to download.

Step 5. Find the message
Scroll down to the audio message. You will see a play button on the message.

Step 6. Open options
If you're using Chrome or a Chrome-based browser, you'll see three vertical dots to the right of the audio message. Click on it to open the drop-down menu. If you are using Firefox or an equivalent, right click on the play button (the triangle).

Step 7. Download the message
Click Download or Save Audio File As. The name of the option will depend on the browser you are using. This allows you to download the voicemail message to your computer in MP4 format.
- If the download does not start right away, select a location and click Save Where OK to launch it.