Just to be clear about what this article is describing, we are not talking about the general differences in neck pickup vs bridge pickup construction and tone. What we are talking about here is how the sound of a given pickup would change if you shifted the same pickup at the same height from the bridge position up to the neck position. It definitely will sound different - you'll get a lot more of the fundamental (the "note" you think about playing when you play) vs. the harmonics (the overtones which are multiples of the frequency of the fundamental that give your note it's tone), but why?
For starters, it would probably be helpful to have a mental model for what the string is doing when you pluck it. It is fixed at the ends (I hope!) by the nut (or the fret you pressed) and bridge, so at those points the string essentially does not move. When you play a note the lowest frequency produced is the fundamental, which is the waveform shown at the top of the below chart. If this was an open strong being plucked, the maximum amplitude of the string would be right over the 12th fret. Since guitar pickups measure string movements, if you had the string vibrating and moved the from one end of the string to the middle, you would hear the output of the fundamental getting louder through your amp.
On the second line of the below chart, we have the 2nd harmonic - half the wavelength and so double the frequency of the fundamental (first line). Each line down the chart shows the next harmonic (3rd, then 4th, etc.), each getting higher and higher in frequency.
So just visually, if you look at the last line (7th harmonic) and find it's point of maximum amplitude closest to the righthand side, if you take that point and go "up" the chart you'll see you are in lower and lower amplitude parts of the lower harmonics. What this means is that if your pickup was placed at that position, you would get relatively more of the higher harmonics than the lower ones, and so you would hear a "brighter" tone.
To demonstrate this I created a basic model for what this looks like as you change position down the neck. The "guitar" is a 25.5" (Fender) scale length guitar, I'm modelling an open string, and I've assumed that for the sake of ease all harmonics have equal amplitude and they all max out at a value of "1". So if all harmonics were maxed at the same position, you'd see a chart of ones below. Note that the "first" harmonic is the fundamental, in case that was confusing.
At 1.5" from the bridge (approximately the Les Paul bridge pickup center distance) you see the exact behavior we predicted above. The fundamental is dwarfed by the higher harmonics, leading to a bright sound.
At 3" from the bridge (approximately the middle Strat pickup position) there is a null moving into the really high harmonics, but note that there is going to be little string energy there - in this simulation all frequencies had equal energy, but in reality the fundamental would have the most, then the 2nd harmonic, then the 3rd etc. (more or less). So still the trend of more mid harmonics holds, and more importantly note how much more of the fundamental you have vs. the 1.5" from bridge example (twice as much).
At 5.5" from the bridge (approximately the Les Paul neck pickup center distance) here you have double again (almost) the fundamental vs. the 3" location. The dropoff in the 3rd, 4th, and 5th harmonics would definitely be audible vs. the bridge or middle positions, which would lead to a less bright and mellower sound. This shifting harmonic mix, along with the differences in neck pickup design and construction, leads to the "neck" sound. If I had to pick one or the other though, I'd say the harmonic mix is the greater driver of the sound difference.
One more thing just for fun. This has been all assuming open string positions, but what about as you fret up the neck? As you play higher and higher up the neck both pickups get more and more of the fundamental (for the neck only to a point, after about the 14th fret you are back to losing fundamental again as you go higher). That is part of why a guitar can sound sweeter and more euphonic the higher you play it. Let's take the 7th fret of a Les Paul and look at the neck position again. Note the strong null on the 3rd harmonic - you basically aren't going to pick up much of it from this string position (you'll get some because the pickup measures more than just one point on the string, but you won't get much). Why does the null on the 3rd harmonic matter? Because odd-order (3rd, 5th, 7th) harmonics have a subjectively harsher sound than even order (2nd, 4th, etc.) harmonics. In this position you get lots of fundamental, 2nd, and 4th harmonics - pure neck pickup bliss :-).