So first, why do we need power regulation or conditioning anyway?...
I just re-read your post and Im glad you mentioned the toaster! The toaster is like most electrical appliances in the world - it isn't very picky about it's power - and we would never notice the difference in performance. All the toaster does is take wall voltage and feed it to the heating element (the portion that glows red). If the wall voltage dips to 100, the power to the element drops proportionally. The toast will toast maybe ~18% slower in this extreme case. Noise on the AC line would have little effect on the element- the element takes a long time to heat up and cool down compared to the speed of the power changes, so it won't care about momentary dips. For example, if a refrigerator kicked in in a poorly wired house, the power may go down to 80V for a tenth of a second. The temperature of the element probably would not change measurably, and the end product (toast) would definitely not suffer.
With audio equipment, we are using our ears to judge the results. We're going to hear and care about sudden dips and changes in wall voltage, unless we regulate the power.
Digital equipment, like a computer, relies on constant voltage levels inside it's microchips to operate. These voltage levels (usually around 3.3 volts DC, or less) dictate the 'ons' and 'offs' at the heart of these devices. If the voltage goes outside a very narrow range, the digital does *not* degrade it's performance - it just ceases to operate.
The L1 has both analog audio portions (preamps, power amps), and digital portions (ToneMatch EQ's, effects in the T1, etc). Both of these portions rely on regulated, constant power to operate well. In the case of the digital portions, it relies on this constant supply of power to operate at all.
The power needed inside the L1 is DC. The amplifiers need around +/- 30V DC, the preamps need around +/- 18V DC, and the DSP's less than 5V.
How's it work?
The L1 uses a low noise switched-mode power supply. These are very light and efficient. Here's an article on how they work: http://en.wikipedia.org/wiki/Switched-mode_power_supply
. In a switched mode supply (or switcher), the DC voltage is created from AC by rapidly switching a transistor on and off. The average of these on and off times, after a lot of filtering, becomes the DC voltage (pictures are in the wikipedia article). Regulation is done by varying the amount of the time the transistor is on and off. The switching occurs very fast, always more than 30,000 times per second. Sometimes much more. Switchers are an abstract concept and there isn't really a good analogy that I can think of for how they work.
For our supplies, there is a lot of extra filtering and tighter than usual regulation to make sure none of the artifacts from the wall power are audible. This is a major part of the product's design, and the focus of a lot of testing.
For some more reading on power supplies in general: http://en.wikipedia.org/wiki/Power_supply
Hope this is helpful and answers your questions -