What does countably many signify? Consider the following definition of discrete Random Variable from the book titled All of Statistics: A Concise Course in Statistical Inference 

X is discrete if it takes countably many values

What the word many says in the definition? Consider the following 

X is discrete if it takes countable set

I hope the above statement is true. Does the word many says that there should be multiple values?
 A: It's standard English grammar.
You would say, for instance, that a set has extremely many elements, or that a function takes remarkably many values (not that that has any formal mathematical meaning). "Countably" works in exactly the same way grammatically as "extremely" and "remarkably".
A: In English, "X-ly many" means the same thing as "an X number of", where X can be a wide variety of adjectives.  So "countably many" means exactly the same thing as "a countable number of".  It doesn't mean there actually literally are "many"; there could be none, or just one (since an empty set or singleton set is still a countable set).
Your proposed phrasing "it takes countable set" is not grammatical English since the noun phrase "countable set" requires a determiner.  If you instead said "it takes a countable set" that would be grammatical but would be unclear: what does it mean for a random value to "take" a set?  The correct way to use a phrasing along these lines would be "it takes a countable set of values".
