On the Play Store page you can find a tutorial video which guides you step-by-step how to export your messages from Windows Phone. If your Windows Phone device hasn’t an SD Card, you can use ‘contacts+message backup’ app to export your messages. If it has SD Card, you can also use ‘Transfer my data’ app. Both apps export a file (.msg from ‘contacts+message backup’ and .vmsg from ‘Transfer my data’).

Simply copy the .vmsg/.msg into your Android device and the app will search for all the backup files. You can select from which numbers you want to import your SMS messages or you can import them all at once.

If you want to import MMS messages or group conversations, try the PRO version, called Import SMS from Windows Phone PRO. Read the description of the app; are the same steps: you copy the exported file(or the PDU files) from the Windows Phone to the Android Phone and you use the Import SMS from Windows Phone PRO app to import all of them.

Thank you very much!

Thank you for the comment Lynn Lucas. Follow the steps below.
1. Sync your contacts (settings > scroll down till you see accounts > tap on Google > tap on the email you want to sync to > tap on 3-dot menu > tap sync now)
2. Backup your SMS using SMS Backup and Restore. Open SMSBackupRestore using a file manager. Copy the XML file to your PC. When you buy a new device copy the XML file to your new device’s SD Card (inside SMSBackupRestore folder)
3. Backup apps using Titanium Backup (for rooted) or My Backup Pro (not rooted).