I agree this bed looks awesome. The frame looks super simple too. I think with the legs you might need some metal "L"Bracket supports on the underside? I'm not 100% sure. I'd like to take the idea to a professional and ask their opinion. Is there any way to get some added support on this from anyone on this website?
I think that if I were building this I would start by making a box of 2X4's set on their side. Since a King bed is about 7ft square, I would make the 2X4 box about 6ft square. I would also run an extra 2X4 across the middle of the bed. With the platform in place the 2X4 frame would be hidden from view by the overhang.
The legs would be bolted to the 2X4's using heavy lag bolts. Put a centre post or leg in as well. I know from experience that a King bed needs extra support up the middle of the mattress.
Now put the platform on top of the 2X4 frame, and stand it all up on the legs.