The dictionary says 生词 means "new words". I know that 词 means "word". Then I concluded that 生 would mean "new" as well as the famous meaning "birth" that it has. But I couldn't find the meaning "new" for the word 生 in the dictionary. Could anyone help me understand how did 生 come to have "new" meaning? I have also come across 生 somewhere else where it meant "production". Again, I couldn't find the meaning "production" for it in the dictionary.
生 does not mean "new". It has the meaning of "unfamiliar" (生疏，不熟悉的）. "生词" literally means unfamiliar word.
生 can be used for "unknown/unfamiliar things" such as 生人(an unknown person), 生面孔 (an unknown face), 陌生(unfamiliar), 陌生人(stranger).
生字 and 生词 mean a word or a phrase that is unknown to the person prior to seeing them or, been taught the first time. In this case, you can think it means "new", a new word, or a new phrase to that person.