In a nutshell: different hands have different requirements.

Your left hand (assuming you are right-handed) has to comfortably reach around the neck to fret the notes. Since the indispensible lowest-string notes (the ones you cannot play on a higher string) are in the lowest position, you want the neck to be reasonably narrow near the headstock - especially on a bass with more than four strings.

Your right hand will ideally want some space between strings, especially if you play slap bass. Moreover, it helps if your 5+ string bass has the same general string spacing at the bridge as a four-string, so the playing experience is similar.

How can we satisfy these contrary requirements? Make the neck narrower at the headstock and wider at the body (the width of the neck in high positions isn't so much of an issue for your left hand, since the crucial notes are going to be easy to reach anyway, being on the highest strings).

In a nutshell: different hands have different requirements.

Your left hand (assuming you are right-handed) has to comfortably reach around the neck to fret the notes. Since the indispensible lowest-string notes (the ones you cannot play on a higher string) are in the lowest position, you want the neck to be reasonably narrow near the headstock. Also, any single-finger double-stops - or barres - are going to be much easier to play if the strings are close together.

Your right hand will want some space between strings, in order to cleanly articulate notes using a variety of techniques without the other strings getting in the way - for example slap bass. A further consideration, as far as bass guitars with more than four strings are concerned, is keeping the same general string spacing at the bridge as a four-string, allowing for a similar playing experience.

How can we satisfy these contrary requirements? Make the neck narrower at the headstock and wider at the body. The left hand is happy, because it doesn't have to reach that far and the right hand is happy, because it has space to do its work without hitting adjacent strings. A neck that widens towards the body end is a standard design that can be found on most stringed instruments with a neck, as Todd Wilcox points out. The width of the neck at the body end is not such a big issue, because the crucial notes are on the highest strings and thus easy to reach regardless of neck-width, while the hard-to-reach notes on the lower strings can be played in lower positions.