Use iCloud backup and restore the backup on a computer
Sync Messages with iCloud, then access messages on a Mac with the Messages app
Use Finder on Mac to back up the iPhone to the computer
Use iTunes on Windows to create an iPhone backup on the computer
Use third-party iPhone data recovery software to export text messages
Use the Messages app on Mac if the iPhone and Mac share the same Apple ID
Save message screenshots to the computer manually
Email or copy important message threads to yourself from the iPhone
Use AirDrop to transfer message screenshots or exports to a Mac
Use a USB cable and a backup/export tool to save messages as PDF, CSV, or TXT