Merry Christmas in Chinese is 圣诞快乐(shèngdàn kuàilè) and its literal meaning is to be happy on the birth of a saint.
圣(shèng) in Chinese is a title of respect for honorific figures or things. For example, Confucius is one of the first 圣人(shèngrén) saints in Chinese history and so is addressed with this honorific title. Another example is that marriage is 神圣的(shénshèng de) sacred.
诞(dàn) means geniture and we usually say 诞生(dànshēng) to be born in Chinese.
Therefore, 圣诞(shèngdàn) refers to the birth of a saint, specifically, Jesus.
快(kuài) means quick or swift in its literal sense, which is used to describe speed. For example, he ran very 快(kuài) fast.
乐(lè) means to be happy or cheerful and is used to describe a positive state of mind . It is usually used with a combination of other words like 快乐(kuàilè) happy, 乐观(lèguān) optimistic, 乐意(lèyì) be willing to etc.
So 快乐(kuàilè) takes on the meaning of 乐(lè), which means to be happy.
As Chinese Christmas greetings, especially in text messages, 生蛋快乐(shēngdàn kuàilè) meaning “lay eggs happily” is popular among young people. This is a play on words as 生蛋(shēngdàn) and 圣诞(shèngdàn) have similar Chinese pronunciation.