1a: Give the legalistic explanations for such a prohibition
1b: Give the abridged response of Shlomo Auerbach, z"tl along with the explanations of 1a
2: Give my concluding thoughts on 1a and 1b, as well as what this means for the greater halachic corpus
Why This Prohibition?
The modern-day poskim have given six reasons for ruling that one cannot turn on or off an object that requires electric power. They go as follows:
1) מוליד. According to Rashi, "creating anything new" was a rabbinic prohibition. Many rabbis have tried to apply this concept to electricity, but as Chacham Tzvi (responsum #92) points out, you cannot extrapolate this principle beyond the explicit situation of applying fragrances to clothing. Furthermore, as Rabbi Michael Broyde points out, "any creative act which is routinely done and undone throughout the day cannot be included in the rabbinic prohibition of creating something new."
2) בונה. The prohibition of building is the most common reason given for this prohibition. Turning a useless wire into a functioning one is seen as analogous to completing a wall. But Auerbach has a significantly better analogy: opening and shutting a door. Doors are constantly open and shut, much like electric items are constantly turned on and off. In halacha, the notion of building has a sense a permanence, something which is lacking in turning on and off an electric object.
3) Turning on an appliance, or ma'keh bepatish (literally "final blow of the hammer"), completes the item (Chazon Ish Orach Chaim 50:9). As Auerbach points out, ma'keh bepatish involves permanence or great effort. Since turning on or off electrical items require neither, this is not the reason for the prohibition.
4) Generation of sparks (Chazon Ish Orach Chaim 50:9). When the ruling was made, a generation of sparks was an issue. First, the sparks are created unintentionally (davar she'eino mitkaven), but more importantly, this argument is technologically moot.
5) Increased fuel consumption (Chashmal Leor Halacha 2:6). Supposedly, the use of electrical appliances leads to an increase in fuel consumption at the power station. Auerbach (Shmirat Shabbat Kehilchata 1:23 n. 137) disagrees for two reasons. One, turning on an electric object would only indirectly cause the increase, which is not forbidden. Two, this is a statistical improbability. Odds are that as you are turning on an item, somebody else is turning off another item, making it immaterial.
6) Heating of metal (Chazon Ish, Orach Chaim 50:9). Turning on an electric item "cooks" the wire, which is prohibited on Shabbat. This is yet another argument that has reached obsolescence because of the advancement in technology.
Auerbach's Concluding Thought
We have gone through each argument that the mainstream traditionalist community uses to justify this prohibition, and as it turns out, none of them muster legitimacy. What I find very telling is Auerbach's conclusion of his own analysis:
"In my opinion there is no prohibition [to use electricity] on Shabbat or Yom Tov... There is no prohibition of ma'keh bepatish or molid... (However, I [Rabbi Auerbach] am afraid that the masses will err and turn on incandescent lights on Shabbat, and thus I do not permit electricity absent great need...) ... This matter requires further analysis. However, the key point in my opinion is that there is no prohibition to use electricity on Shabbat unless the electricity causes a prohibited act like cooking or starting a flame."
We need to keep something in mind. Auerbach was not some liberal, hippie "rabbi" without any credentials. He was a Haredi heavyweight who knew his Torah. What is even better about this statement is that technology has caught up with his main concern: turning on incandescent lights on Shabbat. Now, we have Shabbat switch protectors to prevent Auerbach's primary concern from happening.
Where to Go From Here
As previously stated, from a theoretical standpoint, there is no prohibition of turning on or off electricity. However, there are two obstacles from bringing this from theory to practice. The first is tradition. It doesn't matter that the ruling was made when electricity was merely a luxury item. And in spite of the fact that a hasty halachic ruling was made without looking at the science, they will still preserve it "for the sake of tradition." At least from what my Orthodox friends tell me, the laws can theoretically be changed, especially since this law was created during the Acharonim period [thereby making even easier to nullify], but they will lean on the excuse of "it's a minhag (custom) that has been in our community since the inception of electricity, therefore we'll keep it." To make the point more clear, using the excuse of "it's a minhag" makes it impossible to change anything, thereby making modern Orthodox practice increasingly insipid over time.
But if the ruling is based on a falsity, why keep it? At this point in time, it's safe to say that this is a symptom of the problem in the Orthodox world's overt obsession of codification. Being OCD on codification leads to an overall embrace of "stringency for stringency's sake," and it should go without saying that this trend leads to an increasing intolerance for dissent. If the Orthodox rabbinate is going to continue with its authoritarian, vice-like grip on the lives of Jews, then there is no sense of having a discussion with them in regards to preserving both the letter and the spirit of Shabbat law.
Based on both the science and the halachic analysis of Auerbach, there is no reason to prohibit turning on or off electrical items. This realization, however, does not automatically mean we can use any electric object that we want. We still need to ask ourselves whether a) the usage of this particular electric object will violate other מלאכות, and b) the usage will violate שבות (shvut), which is loosely translated as the "spirit of Shabbat." By doing this, we maintain the integrity of the halachic system, as well as the spirit of Shabbat.
3-3-2017 Addendum: I thought I should put the Conservative movement's take on electricity and Shabbat here. I would argue that it's one of the best teshuvot coming from the Conservative movement.