Remark: The guide below uses t AST Android SMS transfer for Windows version screenshots. The way to finish your task on Mac is almost the same.
To transfer the backup SMS to computer, first launch the AST Android SMS transfer on your computer and connect your Android mobile to computer. Then copy the SMS backup on your computer. On the AST Android SMS transfer panel you will find Import button, click it, then a window will pop up for you to choose the .db files. Choose the .db files you want to view on the computer.
Importance: Because of the vulnerability of the USB cable, make sure to copy the backup SMS files on your computer before importing.