The full explanation is a bit esoteric. Perhaps the most approachable explanation of Godel's proof can be found in Douglas Hofstadter's book "I Am A Strange Loop". Here's my attempt at an analogy using logic and the english language.

Let us say that we hate ambiguity and set out to prove every possible sentence in the English language as a true or untrue statement. Ambitious but doable, no? "Elephants can fly" is false. "Elephants are larger than mosquitoes" is true. Simple. OK, how about: "Using the rules of formal logic, this sentence can not be proven to be true." Uh-oh. If we try to prove this sentence is true, we immediately undermine it. Curiously, the same thing happens if we decide to prove this sentence is false (i.e., it's false that the sentence can not be determined to be true == we can determine that the sentence is true, but that means, by its very text, that it's a true statement that it can't be true). Here is an example of a statement that is "true" (we know in our gut that it's true) but not provable (i.e., trying to use logic to prove this immediately undermines it).

The astute reader may say "Ah ha! The problem is self-reference -- the sentence is talking about itself and that is going to inevitably lead to problems and paradoxes. Let us devise a system of language wherein self-reference is banned." This is precisely what Bertrand Russell and Alfred North Whitehead tried to do in their Principia Mathematica. Self-reference had long been a bugaboo in the field of mathematics and their work was an attempt to establish a complete, consistent mathematical framework wherein all mathematical calculations could be performed but the existence of self-reference was eliminated. Godel, in his famous paper, proved that it was impossible to eliminate self-reference. Again, the reasons why are esoteric and beyond the scope of this text box but I strenuously recommend anyone who finds this to be intriguing to read that Hofstadter book. It is a great examination of Godel's proof and one comes away awed at Godel's brilliance.

The implications of this proof also go far beyond the scope of this comment but are incredibly far reaching in ways both obvious and less so. His incompleteness theorem ranks with Einstein's Theory of General Relativity as one of the greatest and most important discoveries of the 20th Century in my opinion.