This occurs when you enable Messages in iCloud on a new device or after a software update. The system is retrieving your old conversations and attachments from Apple's servers to ensure all your devices show the same message history.
Apple does not provide a single "Download" button for text messages within iOS. To save a readable copy of your chats to a Mac or PC, you must use one of the following methods: Syncing with a Mac apple text download
If the download is stuck, ensure you have a stable Wi-Fi connection and that your device is plugged into power. Sometimes toggling Messages off and back on in your iCloud settings (Settings > [Your Name] > iCloud) can restart the process. 2. How to Download Apple Text Messages to a Computer This occurs when you enable Messages in iCloud