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:
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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.