Why we step on the circle
Circular motion is a fundamental geometric principle in medieval fighting, and the Liechtenauer tradition is no exception. As the author of Codex HS 3227a puts it:
He (Liechtenauer) also means that you should not step straight in with the blows, but from the side at an angle so that you come in from the side where you can reach him easier than from the front. When you strike or thrust at him, he will not be able to defend with other techniques and neither lead it away by changing through as long as the strikes or thrusts are to the man, to the openings to the head and the body with steps and leaps in from the side (Cod.HS.3227a or Hanko Dobringer fechtuch from 1389, translation by David Lindholm and friends, 19V).
The reason that circular motion is so important is simple: it takes advantage of the fact that you can reach further in one direction than in another.
Geometry
If you stand in position and swing your sword through the range of leger available, you will form a lopsided half-ellipse when viewed from above, with you at the center. It stretches in front of you to Langenort at its longest point, and to the side to Schrankhut at its narrowest point. This is your zone of offense. The same ellipse can be imagined for your opponent.
If you made a piece of string the length of the long axis of this ellipse, and tied it to your opponent, you could scribe a circle around him which would show you all the places from which you can strike him. If you drew this circle on the ground around him, you could know precisely where to step with your lead foot in order to hit him; like so—
Notice that both fighters are out of range to hit the other. If Fighter Red were to step directly forward to attack, he would come into the zone of offense of Fighter Green, and possibly be hit. However, because his zone of offense is an ellipse, there are places on Fighter Green's circle where Red's zone of offense intersects Green's, but Green's does not intersect Red's. Therefore, if, when Red came to attack, he stepped onto the circle at the closest convenient location outside Green's zone of offense, he could strike without being struck in return; like so—
Notice that Red's strike will land, while Green's would not. Furthermore, even if he were in range, Green could not land a strike without first getting through Red's well defended centerline.
It is because of these elliptical zones of offense that linear movements forward or back are to be avoided in most circumstances. Passing directly forward will move you into your opponent's zone of offense. Passing on the circle, however (often called a "slope" step), will keep you outside his zone of offense. This doesn't mean you should step as far onto the circle as possible, though. On the contrary, you should step to the nearest possible location, in order to give your man as little time as possible to respond. Smaller steps are better, provided they get you to the circle. Too small, however, and you will remain inside your man's zone of offense when you strike.
Biomechanics
In order to achieve the shortest step, you need to align your body in such a way as to maximize your reach. If you align yourself inefficiently, and cause your reach to be shorter, you will have to make up for it by stepping further, thus wasting time. You will also generate less power, because much of it will be expended in the wrong direction. Because of this, the circle around your opponent is not the only one with which you must concern yourself.
To make an effective strike, you must pivot your body around your stationary foot—thus moving in a circular rotation. When your lead foot lands, your toe should be pointing to where you are striking. If your target moves, the orientation of your toe should follow. (This is impossible if you are not stepping on the ball of your foot, or if you are planting your heel after stepping.) If you do not orientate your toe correctly, the rest of your body will be misaligned as well, and your power generation, reach, and balance will all be poor.
Once your lead foot has landed, your rear foot turns to correct your stance. This rotates your shoulders, pulling your rear shoulder out of your opponent's zone of offense, and further extending your range by moving your lead shoulder forward. At this stage, you are in range to hit him, while he is out of range to hit you. And, even if he could hit you, your centerline is perfectly closed by your own weapon, while his is opened.
Obviously this is assuming that your man does not respond to your strike. If you have stepped correctly, the only possible response available to him is to rotate himself around his lead foot with a smaller motion than yours (because he has less time, being that he is reacting), so as to mimic your own orientation. By doing so he will close his line, and seek to open yours. You are now both in each other's zone of offense, and the krieg can begin.