How to open this electrical junction box?
The fitting is press fit together, and then the melted plastic holds the unit together. DON'T use any metal screws. What has happened is your wire has come loose from the strain relief. If you did anything - attach a strain relief to the backing, push your wire back into the connecting box. Then tie wrap your wires to the strain relief that you attached.

The image below shows only the self adhesive strain relief attached to your cable connector. So insert your cable back into your connector and tie the cable to the strain relief with a wire tie. 

If you pull the unit apart you might have more issues (with looseness etc.)  you could do it your way and pull it apart and maybe wire tie the unit together. My advice is to get a new blanket as these things can be dangerous when they fail.