To answer the how, I have to go back to the day, at post-training tea, I asked him what feeling he has when he performs a technique on someone. His response was to go downstairs to his library and pull out a book by or about Ralph Waldo Emerson, where he had highlighted a passage. In it (going from memory) Emerson describes staying at a farm, whereby he is asked to help take a horse out of a stable. Unable to do so (the horse just pulls back against his efforts) he is surprised by the ability of a child to lead the horse out easily.
The key aspect of this is: The initial destabilisation is only very slight. Consider for a moment a person standing in a stable posture. If, without their feet moving, they move one of their limbs away from their body, or one is moved away, they will be brought into instability. Naturally, our body reflexively corrects for this by moving our feet to bring ourselves back into stability. We do this, in essence, when we walk. We extend a foot forward, taking ourselves off balance, then shift our weight forward to bring ourselves back.
Back to the exercise we did with Katsuhiko Sensei, if you hold a fist out in front of you without moving it, and turn left or right on the spot, your fist will draw an arc. If we consider just the horizontal plane for simplicity, shihonage takes uke's hand in a straight line across their body. If you touch an arc and a line together, from that point they diverge, first only a slight amount, then a great deal more.
Going back to my description of how a person is destabilised if they extend, or have a limb extended away from their body, a technique like shihonage takes them out of stability starting with a very slight instability into a large one (the divergence of the line from the arc). This is the key.
The reason it has to be slight for it to work has to do with how our body reflexively reacts to being destabilised. If you pull or push on someone forcefully, they will, in turn, pull or push back in the OPPOSITE direction. But, if you draw someone very slightly out of stability, they will, reflexively, correct for it by moving their body in the SAME direction. Thus, leading the horse. In essence, along with an aiki age-like feedback loop, it is tricking uke's body into leading itself into great destabilisation. The combination of this with decades of well-developed internal structure, technical skill and timing, has resulted in this seemingly magical ability to perform the techniques without any effort.