Not sure yet what the issue is (assuming you have a 64 bit install), here it does want to remove a whole bunch in synaptic.
(bit different then you but I've other things installed
So for a 64 bit install synaptic only shows the :i386 packages when searched directly.
If you were to click on "Architecture" > arch:amd64 , then find handbrake* you'd come across the proper amd64 packages that install as expected. (see screen), you have to scroll down when in a particular arch, bit of a pita
Or try from command line as a sim - should work fine
If ok then remove the -s from command
sudo apt-get -s install handbrake-gtk handbrake-cli
Not sure whether this is some issue in synaptic or the ppa of handbrake
software-center does show all 4 (:i386 & amd64), screen 2, but what's curious is the :i386 are at the top & shown as the "(default)" ?? - screen 3