Mapper creates a file in the ".raw" format, which contains 16-bit height data. It does not output a .png file. Unreal (as far as I know) can import .raw files.
The .json file is the save file that allows you load the exact same coordinates, output size, etc in case you need to regenerate the heightmap at a later time. This file can be saved in your documents folder, the VCS repo for your game, or anywhere else