Can anyone explain intuitively why increasing the circumference by 1 meter always increases the radius by 15.9cm? I found the mathematical proof, and it is obviously correct. But how can the increase in radius be constant regardless of the starting circumference? With a very small circle, the increase should be huge but with a massive circle, the difference should end up miniscule shouldnt it? After all, the 1 m is getting distributed over a much larger circumference

Let the radius of the sphere be R and the new radius be R', hence

$2\pi R' = 2 \pi R + 1$

or,

$2\pi(R'-R) = 1$

or, the height $R' - R$ is $1/{2\pi} = 15.9 cms.$
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
Because a circle with a circumference of 1m has a radius of 15.92cm.
So whenever you add 15.9cm to the radius of a circle you add 1m to the circumference.

Or whenever you add 1m to the circumference, you add 15,9cm to the radius.

Because $2\pi \times 0.159155 \approx 1$.