Here is the card. As you probably noticed there are eight character cells, arranged in four rows of two characters. The top and bottom rows have the English characters for "Merry Christmas" hidden in Chinese-like characters. The central four characters contain the Chinese text (it reads "Shen dan kuai le") reading from top to bottom and left to right:
Let's look at how the Chinese characters end up on the stamp. Here is the modern Chinese, as you can see it looks rather different since it is adapted for writing with a brush:
The characters are transliterated into the seal script characters with the same meaning:
You can probably see the similarity although the script is much harder to read without training. The "engraving" of the stamp further stylizes the characters. Now you can probably see how they fit into the design.
Real stamps are carved in stone, wood or plastic in special stores. You will see stamps that look like this printed in red ink on official company documents (the real stamp is about 3cm square). Individuals also have their own personal stamps that are used in place of signatures on contracts, for collecting registered mail, and in banks. They are 1 to 2 cm and usually square. Here are mine, one uses the seal script and is for more official purposes (it has a special box), the other is a simpler one for day-to-day use and uses modern Chinese script: