Good question.
First i should point out that all of the planets are only close to the same plane. Usually we define a reference frame defined by the orbit of the earth, then refer to the angle of other planets' orbits relative to that. For instance mercury (with the largest angle of inclination) is about 7 degrees off (i think).
Anyway, as you suggested those are very minor angles.
The reason they are all so close to planar is due to their formation. When a nebula (collection of tiny tiny tiny particles) starts to coalesce into a planetary system - a nucleus has to form that becomes the sun. As particles "accrete" together, increasing the mass, a lot of angular momentum has to be conserved (think of spinning on a swing and pulling your legs in - you speed up), so the whole system starts to spin and flatten out into a disk. Once you have the disk, the center keeps increasing in mass, and smaller bodies start to form further out (which become planets). And that's the in-a-nutshell story of our creation, and why we're so close to a plane.
You can't have an accreting sphere (as apposed to disk), because the rotating particles in different planes will collide and flatten into a disk again - so planets can't form drastically out of the plane.
But it should by possible for a system to capture a larger body out of the plane. If this happened it wouldn't be stable for very long however... so overall the chances of seeing something like this are very slim in a single sun system.
In binary and trinary systems, some much more complex dynamics can come into play... maybe its possible there? (i think it would still be rare)