The 떡 in 떡잎 means "rice cake," and the "잎" means "leaf," so 떡잎 literally means "rice-cake leaf."
My biology textbook defines "cotyledon" as "the first leaf (or leaves) to emerge from a sprouted seed." The reason it says "first leaf (or leaves)" is that some plants sprout one leaf and others two. The plants that sprout one leaf are called "monocots" (short for monocotyledon), and those that sprout two are called "dicots" (short for dicotyledon). The following photo shows the difference between the two:
The Korean name for monocot plants is 외떡잎 식물, which literally means "one rice-cake leaf plants," and the Korean name for dicot plants is 쌍떡잎 식물, which literally means "two rice-cake leaf plants." 외 is a pure Korean prefix that means "just one," and 쌍 (雙) is a Sino-Korean word that means "two," "double," or "a pair." The rice plant, by the way, is a monocot.
Okay, but why do Koreans call the first leaf (or leaves) to emerge from a sprouted seed a "rice-cake leaf (떡잎)"? Well, rice cakes are made from rice, so I suspect that when Korean rice farmers in the past saw rice plants sprouting in their fields, they were so happy and excited that they yelled, "떡이다," anticipating all the delicious rice cakes they would be able to make from their crops of rice. In other words, they saw a baby rice leaf as a future rice cake.
우듬지 is a pure Korean word. The Chinese character meaning 우듬지 is 梢 (초), which is listed under the "tree" radical 木 (목) in Chinese character dictionaries. If you wanted to describe the sound of treetops rustling in the wind using Chinese characters, you would write 梢梢 (초초).
The human nervous system is divided into the central nervous system (중추신경계 中樞神經系) and the peripheral nervous system (말초신경계 末梢神經系). In the Korean name for the peripheral nervous system, the 말 in 말초 (末梢) means "end," so the word that Koreans use to mean "peripheral" (말초 末梢) literally means "the ends of the top branches of trees." Doesn't that make the Korean word for "peripheral" (말초) easier to remember?
The following is a video of treetops rustling in the wind (梢梢).
ANSWER: Because they help make the hard words easier to remember and understand. Plus, if you want to be able to read original documents of the Yi Dynasty (Joseon Kingdom) and of ancient China, then you will need to learn a few thousand Chinese characters and the grammar of literary Chinese, which is somewhat different from the grammar of modern Chinese.
The video below shows animations of how to write the 214 Chinese radicals. Radicals are Chinese characters that are used to categorize the thousands of different characters used to write the Chinese language. They work in a way similar to how words like "mammals, "reptiles," "amphibians," and "bony fishes" work in Biology to categorize all the different species of animals in the world. For example, one of the 214 radicals is the Chinese character 水(수), which means "water," so things related to water, including "ice" (氷 빙) and "a river" (江 강), are listed under the radical 水(수). Another radical is 木(목), which means "tree" or "wood," so things related to trees or wood, including "a plum tree" (李 리) and "timber" (材 재), are listed under the radical 木.
Other characters listed under the radicals meaning "water" and "wood" are not always as obvious as the ones mentioned above. For example, it makes sense that the character meaning "sweat" (汗 한) is listed under the "water" radical, but why is the character meaning "to decide" (決) listed there?
By the way, 氵(수) is a variant form of 水 (수), so either 水 or 氵 will be a component of all the characters listed under the "water" (水) radical. When the 水 form is used as a component of a character, it appears at either the top or bottom of the character, such as in the characters 泉 (천), meaning "a spring," and 畓 (답), meaning "rice paddy." When the variant form 氵is used, it appears as the left-side component of a character, such as in the characters 泣 (읍), meaning "to cry," and 流 (류), meaning to "to flow." One of the reasons some characters have variants is that they help make it easier to write the characters within the imaginary rectangular boxes that Chinese characters are written. Vertically long, thin variants are often used as side components of characters while shorter, flatter variants are used for top or bottom components. And even when there is no variant, the original character itself can be stretched or flattened to fit within the imaginary box. Notice how the 水 (수) character is flattened when it is used as a bottom component in 泉 and as a top component in 畓. That is because you have to fit two characters (白 and 水 in the case of 泉, and 水 and 田 in the case of 畓) within the same size imaginary box that the single character 水 is written. So, since every Chinese character has, at least, one of the 214 radicals as one of its components or is a radical itself, learning to write the radicals correctly is a good way to start learning to write Chinese characters correctly. By the way, it is important to write Chinese characters in the correct stroke order. And you need to practice writing each character over and over again to develop muscle memory. I really like the following video because it shows how to write each of the 214 radicals in both slow motion and at normal speed. It also shows the pronunciation of each character in Chinese and its English and Spanish meanings. Unfortunately, the video does not show the Korean pronunciations, so I have made and posted a chart below the video that shows the Korean pronunciations of the characters in the same sequence that the characters appear in the video. That means the number 10 character in the video will be the number 10 character on my chart. The Korean script above each of my characters tells first the Korean meaning or name of the character and then its Korean pronunciation. The Korean pronunciation is the last syllable in the Korean script above each character. For example, the first Chinese radical is 一, and the script above it reads 한일. That means 한 is the meaning, and 일 is the pronunciation of that character. Since 한 is the pure Korean word for "one," the radical 一 means "one," and its pronunciation in Korean is 일. The Chinese pronunciation is "yī," but unless you want to learn to speak Chinese, you do not need to learn the Chinese pronunciation. If a box on my chart has more than one character in it, then the character on the left side of the box is the main character, and the character or characters to its right is its variant or variants.
Finally, from my own experience, practicing writing Chinese characters is a relaxing exercise, almost like a form of meditation. As you write the same character over and over again to develop muscle memory and improve your writing style, your mind will probably start to relax as your hand takes over the task. I used to practice writing Chinese characters at night before going to bed, often writing the same character over and over 10, 15, or more times, and after doing that for 30 or 40 minutes, I would sleep like a baby.
The video is 1 hour and 50 minutes long, so many will probably not do it all in one setting. I suggest doing the first 15, 20, or 30 minutes of the video every night until the characters almost start writing themselves, and then move on to the next 15, 20, or 30 minute segment of the video. The objective is to train your hand to do the writing while you just listen to the relaxing music playing in the background. Don't worry about putting the characters into words or sentences, yet. Many of the radicals are used as words by themselves, but quite a few are just components of other characters. Nevertheless, if you can train your hand to write the components and learn their meanings, then later you will only have to learn how to put the component pieces together, not learn how to write them.