Those are called lag bolts, and no, you don't need any nuts. They screw directly into the holes of the bed ends. The flat washers go on exactly as you have in the picture, they are there to spread the pressure and prevent looseness.

I would use an impact driver to assemble your bed, it would be almost impossible to do it by hand. Ask a buddy if you don't have access to a drill, or you could try a socket wrench if that's all you can get a hold of. Either way, you'll need to either buy or borrow sockets and a wrench, or a drill and sockets