There are two ways in which the planetary orbits are considered to be especially "regular":
(1) They all go around the Sun in the same direction, in almost exactly the same orbital plane.
(2) Their orbits are almost circular, and they are relatively evenly spaced.
Neither of these "regularities" is required by the laws of physics, or even the Laws of Planetary Motion proposed by Kepler, and verified and modified by Newton. For cometary orbits the orbits can be of any elliptical shape, orientation or size, and even for asteroids, although the orbits all go around the Sun in the same direction, they are neither evenly spaced nor particularly circular, and the orbital planes can vary by as much as 60 degrees. As a result, we can't explain the orbital regularities of the planets by simply saying that is how things are supposed to be. Instead, we need some kind of explanation based on special factors involving the way that they were formed, or have developed since they were formed.
The Rotation of the Solar Nebula
We explain the first regularity by presuming that the planets formed out of a rotating disk of gas and dust called the Solar Nebula, which surrounded the forming Sun 4.5 billion years ago. In the discussion of the origin of the solar system, it was pointed out that if the Sun formed out a cloud of interstellar gas and dust, as the materials of the cloud contracted towards the center, either under their own gravity or some external force, there would be a tendency for any rotational motion of the cloud, no matter how small, to become amplified as things got closer and closer to the axis of rotation, in the same way that an ice skater who is spinning, if they pull their arms in closer to their axis of rotation, spins faster. This would lead, in the case of the Solar Nebula to a gradual flattening of the cloud, so that it became a roughly circular disk, rotating around the forming Sun in the same direction that the Sun itself was rotating. The planets, if they built up from materials inside such a rotating disk, would of course have the same initial motion as the disk itself, and as a result would all be going around the Sun in the same direction and in nearly the same orbital plane.
Of course if this explanation is correct it leads to the question of why the other objects in the Solar System, such as asteroids and comets, DON'T move in this same way, but that is another topic, which will be dealt with elsewhere.
One thing which should be noted is that although the theory of the Solar Nebula explains the motions of the planets, we cannot accept that theory simply because of that. The theory was originally proposed, in the mid-1700's, by Immanuel Kant, specifically BECAUSE the planets orbited the Sun the way they do, so of COURSE it provides a simple explanation of their motions. In order to actually believe the theory we need to compare OTHER predictions of the theory to the facts which we can observe, to see if they are also correct. It is primarily because of the success of this theory in predicting many other aspects of Solar System astronomy that we believe it to be true. However, given the fact that we do believe it to be true, it does provide a very simple explanation of the motions of the planets.
The Shape and Spacing of Planetary Orbits
The nearly circular shape and relatively even spacing of the planetary orbits are related to gravitational interactions between the planets during the latter stages of their formation, and shortly thereafter. There is a tendency for orbits which have commensurate orbital periods (that is, orbital periods which have a common measure, meaning a ratio close to the ratio of two small whole numbers) to be more stable, when dealing with individual objects, than orbits which are not more or less commensurate. If the planetary orbits were less circular and less evenly spaced, when they lap each other (or more accurately, when one with a smaller orbit laps one with a larger orbit) they might be unusually close together, and have a relatively large gravitational interaction.
Over time such interactions could lead to substantial changes in the orbits, making them either more likely to be stable by becoming more evenly spaced and more nearly circular, or making them less stable by becoming more unevenly spaced and less nearly circular. In the former case continued interactions would gradually move the planets into orbits which are more like the ones they now have, and over time their gravitational interactions would tend more and more to keep the orbits relatively stable, instead of changeable. In the latter case the orbits would gradually move into orbits which are less and less stable, producing larger and larger changes, and eventually one or more of the planets involved would end up with a totally different orbit. As far as we know there is no reason why the situation has to end up as it has, but if it ended up the other way, with orbits which are more changeable, and in particular, if the orbit of the Earth happened to be more changeable, then the weather on the planets would be far less uniform, and life as we know it might not have developed as it did, and we might not be here to enjoy it.
(Note that the Galilean moons of Jupiter also have very stable orbits, which are locked to each other with exactly commensurate orbital periods, and in the rings of Saturn and the asteroid belt there are "gaps" which are a result of commensurability interactions with the moons of Saturn in the former case, and the planet Jupiter in the latter case. This will be explored in far more detail at some later date.)
Kirkwood Gaps (after a diagram by Alan Chamberlain, CalTech, NASA/JPL)
(Plot of 157,000 asteroid semi-major axes, in 0.0005 AU increments)
Asteroids tend not to have orbital periods which are commensurate with Jupiter. Very few have orbital periods greater than 1/2 that of Jupiter, and orbital periods shorter than that are selected against, if a small ratio of whole numbers, such as 1/3, 2/5, or 3/7 (shown by vertical lines).