The reason seems obvious at a first glance: To produce different sounds. But really, bridge pickups can be emulated with a low-pass filter.
If so, additional pickups just increase the price of a guitar without affording any advantage.
Is there any significant reason for the additional pickups?
The pickup you'd emulate with a lowpass filter would be the neck pickup since it picks up fewer harmonics. But it can't really be replaced by a low-pass filter since every string would need a different low-pass filter to show a similar composition of harmonics. Also the neck pickup has quite higher gain than the low-pass-filtered bridge pickup and working with lower gain means a higher noise floor. And while the neck position gives a different composition of harmonics for the sustained string operation, short-term action like pickup noise is still present und putting too much of a filter on those takes from the "presence" of the action. Pickup position has most of its effect on the brightness of sustained notes but does not equally affect the onset.
If there was only one pup position, it wouldn't be as simple to blend. With two or three, there are options - admittedly on most Strats all combinations are not available, but they can be, giving a wider sonic choice. Also, more guitars these days have different pups at different positions, giving even more sound choice, and that's before we start on about single coil/ humbuckers, and coil taps.
I've never really experimented with particularly expensive guitars, but the sound of a bridge humbucker (a pickup with two coils and two sets of opposing magnets close together) is loaded with a lot more upper harmonics than the sound of a single-coil neck pickup, which adds sonic interest when playing a lead part, but makes strummed chords sound cluttered; low-pass filtering a humbucker won't help because anything that will take out the higher harmonics of lower strings will also take out the lower harmonics of upper strings; cranking down the tone control on a humbucker doesn't give a clear sound, but instead a sound which is simultaneously muddled and muffled.
At least on the guitars I've played with, the big advantage of having a second single-coil pickup doesn't come from the fact that it picks up sound differently, but rather from the fact that the neck and middle pickups together have anti-phase wiring and magnets so they have the noise-cancelling properties of a humbucker, but the wider space between the associated magnets means that the combination of neck and middle pickups will otherwise sound much like a single-coil pickup.
As to whether it is better to have neck and middle pickups with anti-phase wiring and magnets, or whether one could omit the magnets from the middle pickup and simply use a coil with no magnets for noise cancellation, I don't know. By my understanding of the physics, the noise-cancelling properties of a humbucker stem from the use of coils wound anti-phase, and the harmonically denser sound stems from fact that using two sets of magnets greatly reduces the open-air portion of the magnetic flux path, increasing the sensitivity to variations in string distance. I would think it would be useful for someone to make a pickup that had two coils just like a humbucker but with only a single set of magnets. I would think such a thing should be cheaper than a humbucker, but I've never heard of anyone making one.
Voltage comes from the pickups and the strings. The metal interacts with the magnetic field and disrupts it INDUCING voltage at frequencies the strings vibrate at.
Where the string is placed over the pickup (or vice versa), determines what fundamentlas and harmonics reach full volume or small/no volume. Imagine a pickup directly under the second harmonic's center node. You now have full voltage for the fundamental frequency and as close to zero voltage as makes no difference for the second harmonic. The fourth and sixth will also be similarly of low/no voltage, and 3rd, 5th, would be full voltage.
Knowing this, you'd want the pickup to be offset slightly from center to try to 'capture' all the frequencies at some relatively equal voltages, for a 'full' sound.
On the other hand, say you want a pickup which loses some harmonics, you place it in the spot to lose those harmonics, and place a second one in a position to get/lose the opposite harmonics. Now you have both pickups producing opposite values, and this allows you to select either the former or latter for different tonal qualities, or blend them together to get all of it.
In addition to what others have said, one major difference between the bridge position and neck position is that on the neck position the pickup sees a wider string amplitude. This is an important detail for genres such as heavy metal, where a tight, well defined rhythm is often the goal, and it's much easier to get this type of sound from the bridge position that sees short amplitudes that are easy to mute when necessary.
When you try to achieve the same tight, well defined rhythm on the neck position, you'll find out that:
