Can anyone explain intuitively why increasing the circumference by 1 meter always increases the radius by 15.9cm?

With a small circle, the increase is huge with respect to the small circle. Basically, for a circle of radius 1cm, the 15.9cm increase is a lot ... with respect to the original circle.

To convince yourself, consider this: When you increase the radius of a 1m radius circle by 1m, the radius doubles, but if you do it to a 100m circle, it just increases by a tiny amount relative to the original radius. So in the case of the larger circle, the increase seems much less because the relative increase is less, but in both cases the actual increase is the same.

It's a similar situation with the 15.9 cm increase. The absolute increase is the same, but the relative increase (which is what our brains find easier to imagine) is much smaller