If you ask Rick Astley for a DVD of the movie Up...

If you ask Rick Astley for a DVD of the movie Up, he won’t give it to you because he’s never gonna give you Up. However, by not giving you Up like you asked for it, he’s letting you down. This is known as the Astley paradox.

This joke is based around a pop culture reference, and also a meme culture reference. To understand the paradox requires the listener to know the lyrics to the song "Never Gonna Give You Up" by Rick Astley. The lyrics state that Rick is "Never gonna give you up. Never gonna let you down, etc." In the first part of the song the jokester assumes that Rick was talking about a DVD of the movie Up when he said NGGYU. This is incorrect - the false assumption is based on a pretty awful pun, however it is also a moderately funny pun.

The second part says that he would let you down, which he explicitly denied that he would do in the lyrics to the song, so he is breaking his word. To call that a paradox is silly because the lyrics to the song were not supposed to be taken as logical statements, but instead as passion fueled ramblings.

To give that paradox a name, and declare it to be "known" makes it seem like people have written about this in a serious way, and that it is such a commonly known paradox that it deserves a name. That is just absurd, and that is why this is so funny.

I must admit, this is the best joke that I have read in a long time. 

As a side note, to make people think about the song NGGYU is known as Rick-Rolling, and it is considered to be a good way to tease people on the internet, and is considered mildly annoying. This might also be a mild form of a Rick-Roll.