The download function allows you to download messages as a zip file.
You can find this operation at the bottom of the messages list screen. Once you have selected all the messages that you want to download, just click on it and your download will begin directly.
The zip file will contain received and generated files related to messages and a "messageRecords.csv" file that contains the list of downloaded messages.
Some limits are implemented:
You can only start 5 downloads at a time. Then you have to wait for completing any download before initiating another.
You can request a download of 10000 messages maximum. If you need to download more messages, split the content of your request. For example, filter all messages per periods in time so that they fit in the limit.