As the brake is applied, the ground applies a force directed backward on the tire as shown, causing the fork to flex backward. Problem is, the brake cable is fixed at one end at the brake caliper and at the other end at the cable stop above the headset..
Think “bow and arrow” and imagine the fork between the cantilever bosses and the top of the headset is like the bow, and the cable is like the string. As the fork flexes back due to braking, the cable tightens like the string in the bow, because its two ends – the cable hanger and the brake calipers, have moved further apart. So even though you may have pulled the brake lever carefully enough to modulate it properly, as soon as the pad slows the wheel down, the fork flexes back and tightens the cable, which in turn pulls the pads harder against the rim. This in turn flexes the fork back further, which tightens the cable more, which pulls the pads harder against the rim, and so on.
Eventually, something has to give: Either the tire must slip on the ground, the rider must go over the handlebars, or the pads must break free from the rim. It is the latter that creates the shudder, the pads bind and release, bind and release, each time allowing the fork to flex back and forth and the tire to roll and stop, roll and stop. This is why the problem goes away in mud and wet sand, because the pad can break free smoothly. It is also why smaller pads with more toe-in help.
If the headset is loose, the problem is greater, because the length change between the brake posts and the cable stop atop the headset is greater as the fork moves back when the brake is applied.
Now you can understand why the advent of suspension forks with the cable stop attached to the brake arch bridging between the fork legs improved braking performance on mountain bikes with center-pull cantilever brakes – the arch with the cable stop took the steering tube’s flex out of the equation.