I think it builds it on the system since they have you install g++ and gdb probably for debugging. That's the impression I get, not entirely sure though. So it probably transfers your code via SSH and then runs it.
It's not super clear but I think that step is actually installing g++, etc. on the Linux host (and in reality they probably mean you should apt-get install build-essential and not just g++, gdb, etc. individually). It's having you install openssh-server which wouldn't really make sense on the client/development PC. Also for ARM you'd need a gcc cross compiler installed and I didn't see any mention of that.
the ssh server is indeed for the remote machine/Linux only, we are working on a prerequisites check with possible automatic acquisition of those tools on Linux if you do not have them installed; ARM on x86 or x64 and x86 on x64 it's not a sscenario we currently support, but x86 on x64 would be the first thing we look at
The dependencies mentioned are for the remote Linux machine you connect to from VS. We use ssh from within VS to copy files across and issue the compilation commands with gcc. There isn't any cross compilation right now even for ARM. Most of our testing here has been on Raspberry Pis.
I meant exactly what you said, they install the compiler and tool set on the Linux host, the Windows end just SSH's into it and sends the code / commands. That is my speculation at least.