As others have stated, you have not identified the actual back plate.
That is the actual back plate. Once it is removed you will find that it has a boss that fits a recess in the chuck itself. When you are buying chucks you purchase the chuck and the back plate and they should arrive separated (Exceptions do exist).
You then put the back plate on the spindle and machine it to fit your lathe. This accomplishes two major things, first it means that the chuck is much more likely to have very little run out as you will be cutting the boss exactly for your lathe. To understand, just slap a piece of bar stock in a four jaw chuck and make sure it is not centered. Carefully turn it down. At some point you will have a "perfectly" round piece of stock that is exactly centered with your spindle. You are doing the same thing when you are creating the boss. You are creating a boss that is exactly centered with your spindle.) The second thing you accomplish is that you get to set the tolerances for what is acceptable to you.
To remove the back plate you need to loosen the three screws on the face of the chuck
Once these three screws are removed it is only the fit between the boss on the back plate and the recess in the chuck that keeps them together. That should be a tight fit. Using threaded rod or sacrificial bolts or screws you can screw them as deep as they will go and then tap them gently, in sequence to drive the back plate off.
Next step is to purchase a back plate from a source that already has the right size spindle mount.
If you do not want to purchase such a back plate, you'll need to purchase some stock to make it from scratch:
1) Mount stock in four jaw chuck.
2) Drill and bore to size.
3) Internal thread correctly. (Fitting can be an issue. You might have to dismount your four jaw chuck from the spindle without taking the new back plate out, turn it round and see if it fits on your spindle.
4) Create the mating surface in the bore to match your spindle. This is what will align your new back plate with the spindle. The threads hold the back plate to the spindle, the mating surface is what puts the back plate in the right position.
5) Verify that everything fits, remove four jaw from spindle, turn it around, screw the new back plate onto the spindle. Make sure there is NO sloop. If sloop, start over.
6) Carefully remove the new back plate from the four jaw. Do NOT let the four jaw fall on your ways. Take the new back plate off the spindle with the four jaw and then take the new back plate out of the four jaw.
7) True the outside of the back plate, just because it makes pretty and is easy.
8) Machine boss in new back plate. Make sure it is a tight fit to the recess in the three jaw chuck.
9) Use transfer punches to mark the locations of the three threaded holes you will need to hold the three jaw chuck to the back plate.
10) Remove new back plate from spindle. Drill and tap the three holes.
11) Attach the three jaw chuck to the newly manufactured back plate.
Enjoy high fives and a beer or good cocktail, you've done something cool.