When you adjust a sight you are changing the angle between the sight line and the bore line (axis), which changes where the bullet impacts. That is because the sight line is where you look at the target, but the bore line is the direction in which the bullet is launched. Thus, if you change that angle between the lines by adjusting the sights, but look through the sights at same spot on the target, you have changed the launch angle of the bullet toward the target. That change in launch angle makes a change in the point of impact that is different at every range. That is because the sides of the angle between the sight line and bore line diverge coming away from the sights. In other words, the sides of the angle get further apart as you travel further downrange, so they move the point of impact more at longer range.
It doesn't take a very big change in sight line and bore line angle to move the bullet a significant distance at most common shooting ranges. For that reason, a very small measure of angle needs to be used to adjust sights. The moa (1/60 of a degree) is the conventional choice. We could have used milliradians (as mil dot sight reticules do) and it would have made more sense mathematically and allowed a common unit system to be used for both rifle and artillery, but moa is the historic unit of measure of the adjustment angle for all small arms sights, so it is the unit our sight settings use.
The illustration below uses an exaggerated angle to provide the distance measures. As I said, above, the sides of the 1 moa angle get further apart the further you go forward of the gun. By the time you get to 100 yards the are very close to pi/3 inches apart (1.0472… inches). That means that at 100 yards a 1 moa change in sight setting will move the point of impact pi/3 inches. If the target was at 200 yards, that same 1 moa sight adjustment changes the point of impact 2×pi/3 (2.0944…inches); by the time you get to 300 yards that same 1 moa adjustment moves the point of impact pi inches (3.15159… inches); etc., etc.
The above is why the adjustment is often said to move the point of impact 1.05" per hundred yards of distance from the target. 1.05" is close enough to the exact number that the shooter won't ever see the difference. It's less than 0.03" of error at 1000 yards; too small to hold the rifle to, and even if you could, it is too small not to be overwhelmed by shot-to-shot velocity variation and atmospheric irregularities.
Note that if you rotate the sight on its axis, the sides of the angle describe a cone, with a circle 1 moa across on the target. When you hear people say they have a 1 moa group, the shots staying inside that circle is what they mean.
Bryan Litz, in his book, Applied Ballistics for Long Range Shooters, says a lot of sights fail to move the sight line the exact number of moa per click the makers claim. Some adjustment mechanisms and optics don't move the same amount per click either side of the center of the adjustment range. Some don't move it the same amount per click near the center that they do near the ends of the range. For a small adjustment you don't notice the error, but for that 30 moa shift needed for 1000 yards, the cumulative error can be pretty far off. There is also the problem with iron sights that are the same, but placed on different barrel lengths where the front and rear sights are different distances apart.
Litz recommends strongly that you measure what your sights do. He sets up a target with a straight line about a yard tall vertically above a bull. You can make your own by putting a standard target at the bottom of a tall cardboard backer and just drawing the line on it with a yard stick. You zero the rifle on another target. When you have that, you put up your tall target and fire a group into the bull. Adjust the sights up 1/4 of the total adjustment range and fire another group. Repeat at half, three quarters, and full vertical adjustment. Find the centers of the groups and measure how far apart they are. That will show you what your adjustments are doing. Divide the distance between each group by pi/3 to see how many moa the group actually moved. Divide the result by the number of clicks of adjustment you used to see how many moa each click of your sights really is in that part of the adjustment range.
In the above text you'll note I said the separation was "about" pi/3" per hundred yards. For the sake of mathematical exactness, the target is a plane and pi/3 is the length of the 1 moa arc at a radius equal to the range, and is not the straight line length that subtends that angle at 100 yards. However, the angle is so small, the arc doesn't have much bend over that short distance, so the difference in its length and the length of a straight line across the diameter of the sides of the angle, is only about 0.00000000397 inches at one hundred yards, and is, again, ignorable. Even elevating a sight 30 moa to shoot at 1000 yards, the error is only about a ten thousandth of an inch. Nothing the gun or shooter can detect.