Just sync your iPhone to iTunes on Mac. And note that the backup is saved as .SQLITE_DB files, you cannot get the files to view directly. So you can extract your iPhone backup file to Mac as excel files.
Make sure to transfer the data from your iPhone correctly!
Actually, iTunes is good to backup iPhone files. But some media files cannot be backed up because of their large sizes. Therefore, I like to use third party tool to help me do that. You can search the related tool on the Internet.
Download iPhone SMS Backup for Mac or PC here. Once you install downloaded software in your computer, it is free to use to preview sms or text messages. Very handy tool to export sms for any legal purpose or to keep them as Excel file
iphone sms backup