At the beginning, I want to say that I am a native speaker and love Chinese, but I am not on a research level. It is welcomed that anyone can make comments and supply more info to my answer.
Introduction
First, Wikipedia (see the link provided by Krazer) is good start to get some background knowledge why we have simplified characters. After the found of the new China (People's Republic of China, P.R.C), the country was out of turmoils and wars and the government decided to simplify the characters (note that before P.R.C, there were some proposals and pratices on simplification). I think one main reason was to allow more people to get well educated, since simplified characters were easier to learn. However, the government didn't make a long-term preparation and thus some characters were not simplified in a "logic" or "scientific" way.
Main Text
“叶” is a typical example of a character that was not simplified "properly". So how did characters get simplified? As far as I know, unfortunately, the government didn't issue any official document to explain this. I searched this topic on the internet and I found some useful info, one of them is a blog, which I think has logic and acceptable explanations.
I will explain more with my own knowledge:
古 (the original characters): One example is "从" and "從". In 《说文》(a famous old dictionary),"从" is explained to be the orignal form of “從". "从" means "to follow" and it's intuitive (one 人 – person – follows another) so "从" is selected.
俗 (the informal characters). In ancient times, few people got well educated and of course, characters were developing too, so there were some "informal" characters, which were popular among not-well-educated people. Two examples are "泪" and "淚". In 《字彙》(a dictionary written in Ming Dynasty), "泪" is explained as the informal form of "淚". "泪" is intuitive, because the left part refers to water, "目" means the eye, and "泪" means tears. It's "reasonable" to use “泪" as the simplified form.
异体字 (characters with the same sound and meaning, but different shape). Example: "凶" and "兇". Both have the same pinyin (xiōng) and same meaning (danger), so "凶" is selected because it has less strokes. Also, "凶" is intuitive: isn't it like a man dropped in a trap?
闲置的旧字 (characters used under very rare conditions; they won't appear in most cases). This is how "叶" is simplified. Actually, even now, "叶" has another pronunciation and meaning, if you consult it in a dictionary (higher-level) and this is actually the original “叶” in classic chinese. It's "叶 (xié)" as in the word "叶韵" (yùn), which means that some characters have similar or the same vowel so the pronunciations of them sound harmonious ( when rhyming). You can find this phenomenon in classic poems and modern lyrics.
However, this meaning is very very rarely used and we would say “押(yā)韵" rather than “叶韵" nowadays. 叶 (xié) sounds similar with 葉 (yè) — remember this happened in ancient times —, so some not-well-educated people used 叶 instead of 葉 by mistake, and this mistake was "admitted" in simplification. Another example is "听" and "聽"(tīng) which means "to hear". "听" (yīn) means "laugh" as in "听然" in Classic Chinese originally, but this meaning is generally expressed by other words like "哂" or "莞尔" in classic Chinese, so this character is borrowed to express "聽" because of similar sound, even "to hear" has nothing to do with the radical "口" in "听".
草书楷化 (simplification from Cao to Kai). This will explain how "单" and "严" came out. "草书" is a hand-writing font. Have you heard of Chinese Calligraphy? 草(草率) implies that characters are written very casually, so characters are difficult to read but fast to write.楷(kăi, paragon）书 is another font in formal writing. This principle is used to simplify a character so that the shape looks like its shape in 草书. Below are some pictures to show.
Characters are 单, 书, 为, 学, 严, 专 in traditional form, from left to right and the traditional forms are: 單, 書, 為, 學, 嚴, 專. (Remember that 草书 has several schools so the shapes vary somehow. Examples below are based on the font of 黎凡草书 installed on my PC):
同音字 (characters with the same pronunciation). Select a character with the same pronunciation to replace a complex one. Example: "后" and "後". "后" is really a character used in Classic Chinese, meaning “the queen" or "the formal wife of the emperor", and “後" means "to be after something, some time point". "后" is selected as the simplified version of "後" and thus has the meaning "after". This also brings problems: when people want to use traditonal characters in some cases, they may use wrong characters. E.g. the year of 2009 was roughly the year of “己丑" in Chinese Lunar Calendar (known as "Sexagenary cycle").
During the spring festival of that year, I found "己醜年" in a website. I understood the site stuff wanted to express the year in a classic way, but they were absolutely wrong. "醜" means “ugly, not beautiful” and was simplified into “丑“. "丑" is really a character used in classic Chinese as an element of the Sexagenary cycle.After the simplificaion, “丑" got two meanings, that's right in simplified chinese. However, you can only write "己丑" in traditional form, and "醜" is incorrect.
Ok, there are some other cases, but I think it's enough now, because this answer is not written as research paper. It's time consuming because I have to consult some characters in 《康熙字典》(a very prestigious dictionary written under the order of the emperor ”康熙" in Qing Dynasty).
At last, I want to say that language is not like Math or Physics, and it's hard to find "proof" or "theory" for some phenomena. Perhaps I can't explain why we speak that way, however, it's the way we speak. Take it easy.
Hope this helps.
