To remove the old handle:
Once the full handle is removed from the door, you need to remove the metal lock casing.
There is a small black plastic catch at the base of the metal casing that holds it into the housing.
Just get a small screw driver into this to wedge the metal housing toward the top.
This allows it to be removed after it clears the black catch.
Once that is out you should be able to remove the handle (you may need to bend or break the small spring if it is still intact).
To replace the new handle it should just push into the same section. When it is in far enough the spring will click into place. Replace the metal housing until it catches under the black lower catch.