Back up the iPhone to your PC with the Apple Devices app or iTunes, then open that backup in OpenExtract, a free Windows app, and export your conversations as PDF, HTML, CSV or plain text files. You can save one conversation, a handful, or all of them at once.
Windows has no built-in way to copy texts off an iPhone, and the backup Apple's software makes is a folder of files with scrambled names that you can't just click through in File Explorer. OpenExtract reads that backup and turns the conversations into ordinary files you can keep, search and print.
Windows needs Apple's software to talk to an iPhone over a cable. Get the Apple Devices app or iTunes from the Microsoft Store, or iTunes from Apple's website. You may already have one of them.
Connect the iPhone with a cable and tap Trust on the phone if it asks. In Apple Devices or iTunes, select the iPhone and choose to back up to this computer. Ticking Encrypt local backup is optional; if you do, write the password down. Then click Back Up Now. OpenExtract can also make the backup itself from its start screen, once Apple's software is installed.
Download it from the Windows download page and run the installer. The installer is signed by Charles West, the developer. If SmartScreen shows a notice for a new app, choose More info, then Run anyway.
On OpenExtract's start screen, choose to open a backup folder, and pick the Backup folder, usually %USERPROFILE%\Apple\MobileSync\Backup (the Apple Devices app, or iTunes from the Microsoft Store) or %APPDATA%\Apple Computer\MobileSync\Backup (iTunes downloaded from Apple's website). Type the backup password if it asks.
Open Messages and click a conversation. Click Save conversation at the top right, choose PDF, HTML, CSV or plain text, and click Save. Pick a folder, and the file is named after the conversation and its dates.
Click Select above the conversation list, tick the ones you want or click Select all, then click the button that shows how many you ticked. Pick a format and, if you ticked more than one, whether you want one file each or everything in one file. Then save them and pick a folder. For one spreadsheet of every message, click Select all, then choose CSV and everything in one file.
Do I need iTunes? You need either iTunes or the Apple Devices app to make the backup, because Windows needs Apple's software to connect to the iPhone. After that, OpenExtract does the rest.
Do I need the internet? No. Making the backup and exporting the messages both happen on your PC.
Where do the files go? Into whichever folder you pick when you export. Nothing is saved anywhere else.
Does it get iMessages as well as SMS? Yes, iMessage, SMS and group chats.
Does anything get uploaded? No. OpenExtract reads the backup on your own computer and doesn't send it anywhere. It goes online only when you let it: to check for a new version, if you allow it (you can turn that off in Settings); to send a problem report or a thank-you, when you click Send; and on Windows, to install Apple's free Apple Devices app from Microsoft, if you ask it to, for backing up an iPhone. Nothing from inside your backup goes with any of them.
Is it really free? Yes. OpenExtract is open source under the MIT license, with no paid tier, no account and no strings attached. The code is on GitHub.
Free, open source, and the backup never leaves your computer.