Just a simple explanation...
A pot is just a variable resistor. It regulates how much signal is bled off to ground. High frequencies are bled to ground faster than lows.
By changing under spec pots, to pots that are on, or above spec, the tone will brighten. This is a good thing if the tone is muddy sounding, but not if it's bright already. The higher the k-ohm value, the brighter the tone.
The rule of thumb is that single coils use 250k-ohm pots, and humbuckers use 500k-ohm pots. In either case, if the pots actually read well below the rated k-ohm rating, the tone can sound dull, or muted. With so many pickups out there that are all over the board on resistance values, tailoring your pots to your pickups is important.
A couple cases in point: I have a set of Rio Grande Genuine Texas/Texas BBQ humbuckers in one of my guitars. I know from experience (and the experiences of others), that these pickups need pots that
read at least 500k-ohm or they will sound dark, dull, and muted. In a particularly dark sounding guitar, some have even gone to 1-meg pots to compensate for this.
Another example is with overwound single coils. Many of these pickups are wound well above the normal k-ohm range of std., or vintage single coils. I have some that read in the 7k-ohm to 8k-ohm range, which is in PAF humbucker range resistance wise. And there are some out there even higher. These pickups don't sound good unless the pots are in the 275k-ohm to 300k-ohm range. Ever heard someone say that they don't like Fender Texas Special single coil pickups because they're too dark or muddy sounding? I'll bet they never actually checked the resistance on their pots, nor replaced them with some of suitable resistance.
Part of the fun of re-wiring a guitar is in figuring out what you've got, and how to go about changing it to get the tone you're looking for.
Here's a great tutorial on guitar wiring from Stew-Mac that explains (among other things) how pots work in a guitar circuit:
http://www.stewmac.com/freeinfo/i-4000/