I've noticed in many basses (not sure if this happens in all of them), the top of the neck starts at a certain width and towards the bottom, it is wider. Here is an example:
This happens to my double bass as well. What is the reason for this?
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The same question for strings in general is discussed here. Here are a few good quotes from that discussion:
The truth is that you want strength at the heel of the neck, you want slimness where your fingers need to go the most, and you don't want a baseball bat for your neck
Then there is the matter of the amplitude of a vibrating string. As you get closer to the 12th fret (midpoint of the string), you need to be sure that the strings are sufficiently far apart that they don't interfere with each other.
and my personal favorite:
A tapered neck will ensure more even levels of stress within the neck. A parallel neck will have higher relative stress levels nearer the body than near the nut. Since the relation betweeen stress and strain is constant, a tapered neck will deform more uniformly along its length under the load from the strings; a parallel neck would tend to bend from the heel. So the amount of taper will affect the shape the neck deforms to under the string load.
The last one implies that the taper of the neck affects how the neck relief (the bow of the neck to keep the strings from buzzing on the frets) is distributed. That combined with the truss rod channel shape allows the luthier to design for a fairly particular relief profile.
It does look like the best answer is probably "all of the above", and by "all" I mean to include Faza's answer regarding playability.
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.