Messages
Why does my phone say it’s downloading my messages from the cloud
iPhone XR, iOS 15
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You have turned messages off in iPhone , however iPhone is synced with Mac ( signed in with same Apple ID and password on the same network ) and in Mac the messages box is turned as on , as soon as you turn messages box in iPhone the messages from iCloud will start syncing .
Set up Messages on iPhone - Apple Support
When you turn on Messages in iCloud, every message you send and receive on your iPhone is saved in iCloud. And, when you sign in with the same Apple ID on a new device that also has Messages in iCloud turned on, all your conversations show up there automatically.
Go to Settings > [your name] > iCloud, then turn on Messages (if it’s not already turned on).
Read Change iMessage preferences in Messages on Mac - Apple Support
Access your messages on all your Apple devices - Apple Support