Why there is no "the" in the second case? For me both of this cases look identical.
His father gave him a bag full of nails. (...) On the first day he hammered in 37 nails, but getting the(1) nails into the fence was not easy. As the days went by, he was hammering in less nails, and within weeks he was able to refrain from getting angry and from hammering
the(2) nails.
"The" is used to signify a noun is the same as one previously mentioned in conversation, or the same as an instance of one directly observable (or obvious) by both parties in the conversation.
On the first day he hammered in 37 nails
This is the first mention of nails which establishes a context. (The would only be used here if there was a previous conversation or the people in the conversation can physically see the nails.)
but getting the(1) nails into the fence was not easy
We say the nails if we are talking about the same 37 nails just mentioned. If we don't specifically mean these nails, but any nails, the would be omitted.
... within weeks he was able to refrain from getting angry and from hammering the(2) nails.
Again, our established context with "nails" is the 37 nails mentioned previously. We would use the if we were still talking about those same 37 nails. Unless you really mean to express that he's spent weeks hammering the same 37 nails, the would be omitted since you really mean "any nails" or "nails in general."
The first "the" refers to "37 nails". The second "the" is incorrect because it doesn't refer to anything in the sentence. If the second "the" still referred to "37 nails", it would not be correct because the sentence says he hammered in less nails as the days went by.
It's that way because the author wanted it that way. Either, neither, or both of the "cases" could be written without the definite article. That is what the definite article does, it makes a definite reference. Not using the definite article with a plural count noun (such as nails) makes an indefinite reference.
There are 4 places where the author can either use the definite article before nails or not use it:
His father gave him a bag full of A nails. (...) On the first day he hammered in B 37 nails, but getting C nails into the fence was not easy. As the days went by, he was hammering in less nails, and within weeks he was able to refrain from getting angry and from hammering D nails.
You would not use the with less nails because less already makes this an indefinite reference. In any of the other four instances, you can use the if you want to make a definite reference. And not use the when you don't want to make a definite reference.
So the usage depends on what kind of reference the author wants to make. Authors make a definite reference most often when they expect you to know which 'things' they are talking about. Note that this expectation does not have to be correct. And this is not the only times that authors can make a definite reference.
Of course, making a definite reference and an indefinite reference changes how we understand nails, so the meaning of the overall passage will change accordingly.
