Could be that they liked them and want to remember them. Or they want to use them for something in the future--like a court case. All depends on context.
I have been known to save very sweet texts from my significant other. Just because I like re-reading them because they made me smile.
They want to be able to read them whenever they want.
Saving old text messages represent that those messages contain some important information or the information about the person u really like so thats why u save them! In fact I save my messages on these basis only!
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