It is actually not, or at least is mostly not, the radiation pressure. The radiation pressure from the x-rays in ICF is not nearly sufficient to ignite the fuel. Rather, it is the ablation pressure from the outer "shell" of the target, which is very rapidly heated by the x-rays. The rapid heating, of course, produces a very large change in internal energy and therefore a large pressure; the outer layer of the target rapidly expands outwards, in the direction of least resistance, producing "recoil" motion in the form of a shockwave directed towards the center of the target.

ablation provides much higher pressures than direct radiation pressure