How do they derive at this answer when the force is lateral?

The key is to imagine that the frame is pinned at the two ground points A & D. And the connections at B and C are hinged. So as the force pushes at point B the frame wants to fall to the left. So the force to resist that 70 kips has to equal it or it will be out of balance. Half of the force is resisted through each vertical leg since they are equal members in terms of size and connection. So 35 kips per leg. Since the force is pushing from the right. The left leg has to resist the force at point B that makes the frame want to collapse. It has to essentially push back but it is pinned at the ground so the force is a combination of the resistance plus the height. So the resisting force travels up from the ground from D to C and then across from C to B. So the resisting force in that leg(C-D) is half the 70 kips x the height of the column. So 35 x 17 or 595.