You can just follow this tutorial about how to download text messages from iPhone to computer step by step. Not only text messages can be transferred, but also MMS messages, SMS messages, iMessages and Messages attachment can be exported or downloaded to computer.
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How to Transfer Text Messages from Old iPhone to New iPhone.
If you have bought a new iPhone 7,and use old iPhone 5s/5/4s before,you can use this phone transfer to transfer any or all your SMS/iMessage messages to computer or new iPhone 7, as well as transfer photos in Camera...