Actually, sms messages are usually saved in the Android phone's internal memory and sim card. To save Android sms to computer, we need use a pro Android mobile manager tool to help us get access to them and export them to compute for backup.
With Samsung Backup software,you’ll be able to copy SMS from android to computer.In Addition,this wonderful program allow to transfer various of data between mobile phone and computer, including photos,contacts,videos,music,document,etc.
Under this situation, I would like to recommend this Coolmuster Android Assistant program to meet all users' needs. With it, you'll have the ability to export any types of files on your phone like text messages, contacts, apps, photos, video, etc. to your computer and save android phone messages as .csv format.
You can use a third party mobile manager app to transfer your Android messages to the computer. There are many available apps that can do so. Just download the app on your PC and mobile, connect phone to the computer and select contacts to transfer. It is very easy,and is what most mobile manager software works, hope it can solve your doubts.