save iphone messages to imac

The crude, no cost way would be to do a screen capture on your iPhone and then send that image (via email) to your iMac.

How to take a screenshot on your iPhone, iPad, and iPod touch - Apple Support

There are more elaborate solutions, but they require a third-party solution. For example:

http://www.iphonetextmessages.com/

(I have not used and can't vouch for any of them.)

One way is to get an app that will copy content from your phone to your Mac. TouchCopy is one that is popular.

You can also set up your phone to reproduce messages on your Mac if they are connected to the same iCloud ID. On the phone, go to Settings/Messages - Text Message Forwarding, and select your Mac. This is not a sync; if you delete messages from your computer or your phone they will not be deleted from the other device.