There may be a more "technical" answer, but I believe it's just because of the way the guitar is tuned (E A D G B E). If you look at it in terms of frets (half step intervals), not notes, you see that everything is lined up the same, just "off" by 5 frets each string. Everything shifts to the left by 5 frets as you go down the strings.

D:11-12-13-14-15-16-ect
A: 6--7--8--9-10-11-12-13-14-15--16--17-18
E: 1--2--3--4--5--6--7--8--9--10--11--12--13

Take "11" for example. Every 11 is the same pitch as the other 11's on the other strings. There are 3 different "11's" here, but they are exactly the same. No difference in octaves or pitch, so there are no intervals between them. Open A would be 5, which is the "5" on the E string.

I'm not sure if the right way to look at is is through intervals, like you are trying to do. Since you aren't going to a different octave. You are going to the exact same note/pitch so it isn't really an interval thing. It's just a pattern to find one note in another place. A shortcut.

Octaves on the other hand are an interval thing. One pattern you may be familiar with to find an octave is to go down 2 strings, up 2 frets. So start with C on the 3rd fret of the A string, go down 2 strings to the G string, then right 2 frets to the 5th fret, and you land on C again. Same note but different OCTAVE. You can look at this with intervals because you now have a distance of A to B between your 2 notes (look at it as "low C" and "high C"). There are notes between the two different C's.

There are intervals between octaves, but not between notes of the exact same pitch, like in the pattern you originally spoke of. That brings you to the exact same note as opposed to the same note, but in a different octave. So there are no intervals between the 2.

EDIT:
If you are looking more for the reason why the guitar is layed out the way it is, then that's a whole different thing. EADGBE is known as "Standard tuning". The reason those notes were picked has to do with what Chris mentioned:
"The purpose of this is to be able to play a lot of notes (or a whole scale in this case) in one fretboard position. "

This tuning is used most often because it allowes for easy finger placement for many chords. Ever wonder why the B string is off-set and you have to tune it to the 4th fret of the string above it, instead of the 5th like every other string? Tune it the same as you would the other strings and it becomes C, then the E string becomes F. Try fingering a chord like that! Bar chords get thrown out the window since you would now be barring the wrong notes on the last 2 strings you just changed. All chords you play using the E and B strings would require you to shift your fingers around in most likely uncomfortable positions.

Have a quick look at the wiki on tuning:
http://en.wikipedia.org/wiki/Guitar_tunings#Standard_tuning

As you can see there are a different ways to tune (drop tuning would be the next most used I think). The whole purpose of tuning different ways is to make things easier to play with your left hand.