Before you start: Messages in iCloud. Apple says a backup made on a computer leaves out iMessages and texts that already sync to iCloud. If Messages in iCloud is on, the backup may hold few or no conversations, and BackupPaper will report that the backup has none. The setting is in the iPhone's Settings under your name and iCloud; read Apple's guide to Messages in iCloud before you change it.
Step one: make a backup on the PC. Install Apple Devices from the Microsoft Store, or use iTunes if it is already on the PC. Connect the iPhone, enter its passcode, trust the computer if the phone asks, and select the phone in the app. Choose to back up to this computer, then select Back Up Now. Encrypt local backup is optional; if you turn it on, BackupPaper will ask for the same password, so keep a note of it somewhere safe.
Step two: let it finish. A first backup can take a while. When it is done, the backup sits in %USERPROFILE%\Apple\MobileSync\Backup for Apple Devices and iTunes from the Microsoft Store, or in %APPDATA%\Apple Computer\MobileSync\Backup for iTunes from Apple's website, inside a folder with a long coded name.
Step three: open it in BackupPaper. Choose that folder and type the password if the backup is encrypted. BackupPaper lists the conversations, newest first, with message counts and last dates. From here on the phone is not needed, so you can unplug it.
Step four: export. Pick a conversation and choose Export PDF for a document to read and print, or Export CSV for a spreadsheet. Name the file in the Windows Save dialog. BackupPaper refuses to save inside the backup folder, so choose Documents or another place you will find again. Each export holds one conversation; repeat for every thread you want.
Step five: note the SHA-256. After each export BackupPaper shows the SHA-256 of the file it wrote. Run certutil -hashfile on the file later and you get the same value as long as the file has not changed. It is a fingerprint of the exported file, not proof about the messages themselves.
What does not come across. Photos and videos stay in the backup; the export names attachments but does not include them. Only the Messages app is read, so WhatsApp, Signal and other apps are left out. BackupPaper shows the messages the backup holds and does not bring back deleted ones.
Keeping the backup. The backup is still a useful copy of the phone. BackupPaper opened it read-only, so Apple Devices or iTunes can restore it later. If you need the disk space back, export every conversation you want before you delete it.