The grammar of the sentence is a bit hard to follow. When I am presenting this paradox to friends (I have interesting friends), I hand them a piece of paper with the following words on it:

Take another piece of paper and copy these words:

"Take another piece of paper and copy these words: "QQQ" Then replace the three consecutive capital letters with another copy of those words. The resulting paragraph will make a false claim."

Then replace the three consecutive capital letters with another copy of those words. The resulting paragraph will make a false claim.

I urge you to carry out the task. You should wind up with a paper that has the exact same words on it as the paper I gave you.

If you believe that the statement on my paper is true, then you should believe that the statement on your paper is false, and vice versa. Yet they are the same statement! Assuming that you think truth or falsehood is a property of grammatical sentences, independent of where they are written, this should bother you. Moreover, unlike the standard liar paradox, the paper I gave never talks about itself, it only talks about a message you will write on some other piece of paper (which does not, in turn, talk about the original message) when you perform some simple typographical operations.

Quine constructed this example to demonstrate the sort of subtleties that come up in order to invent a mathematical formalism that can talk about truth, and can talk about manipulating symbols, without bringing in the liar paradox. (To learn how this problem is solved, take a course on mathematical logic and Goedel's theorem.)