Well, as we know that the memory of your Android phone where all text messages are stored is always limited. If the capacity of the inbox of your Android phone reached a certain degree, you cannot receive new messages any more unless you delete some old ones. However, many messages are so important that you want to keep forever, so what will you do to solve this kind of problem? Then I suggest you to make a copy of text messages on your computer before deleting them from your phone. In fact, you may need to export SMS messages from Android phone to computer in many situations.

While, we all know that we are not able to transfer text messages or contacts from Android phone to PC unless we turn to a third-party program for help. Then you can choose Coolmuster Android Assistant to help, it's easy to use.
Step 1:Connect Android Phone to pc
The first thing you need to do is to download and install the Android contacts manager on your pc and launch it. The next is connect your Android phone to pc via usb cable. Your device will appear on the screen. The manager would recognise your android device automatically.

Notice: If this is the first time you connect android phone to the problem on computer, you should enable USB debugging first, and install a USB driver on your computer, then the app would detect your android phone automatically.

Step 2:Select sms and Start Export
You can find many icon on the left, find "Contacts" and click it, all then go to "SMS", then all sms on your android phone will be listed on the right. Then select and check you want to transfer, at last click "Export" to start the work.

That's all!
http://www.coolmuster.com/android/transfer-text-messages-from-phone-to-computer.html