How to mount floating loft stairs to the wall
Well, the boxes themselves appear to be structurally adequate to carry human weight. The 3-4 inch lap between the boxes also provides some stiffness and weight bearing. The trick would be in fastening them to the walls in such a way as to withstand the large tensile (pulling) forces. A few ideas come to mind:


Decorative or hidden angle brackets, mounted to the wall and the upper surface of each step.
Dual-thread bolts, with lag threads into the wall framing, and machine thread on the extension, which would need to pass through most of the upper step panel with washer and nut applied behind the face trim. 
Direct framing attachment, where the stair components are fitted around the wall framing and anchored to it. In this case, insulation and wall paneling would be fit around the stair components. 


However it's done, the individual boxes need to be well-built, using an appropriate adhesive, quality sheet goods, and mechanical fasteners where possible.