Here's the simple (i.e. less than rigorous) version. It applies to a elementary number theory (ENT). I'll skip the formal definitions, but ENT is basically the theory of the nonnegative integers.
ENT is built up from a number of axioms. If an expression follows from an axiom or a theorem, then it is said to be a theorem of ENT, and the list of expressions it follows from are its proof. To differentiate mathematical expressions from ENT expressions, I will write ENT expressions in bold type.
Now, we can write any expression or proof in ENT as a number, called the Godel number of the expression. I won't explain how to do this, but it is not difficult.
Since we can do this we can define a relation P(m,n) which is true for all numbers m and n, where m is the Godel number of an expression A and n is the Godel number of the proof of A. (Still with me?) I won't explain how, but we can write an expression P(x1,x2) such that P is provable in ENT if P(x1,x2) is true and not P is provable in ENT if P(x1,x2) is false.
Got that? Then let r be the Godel number of the expression for all x2, not P. This expression will be provable in ENT if the expression x1 is not provable in ENT. Note that r is a Godel number. Now, here's the clever bit: let E be the expression for all x2, not P(r|x1). (There is nothing mysterious about P(r|x1). It just means P with x1 replaced by r.)
Why is this clever? Think about what E means. Well, E means "E has no proof in ENT". Is this true? If E is false, then E has a proof, so E is true. But that is ridiculous, so E must be true.
This might seem like a paradox because we have just proven E. This is not a paradox however, because we have proven E in English, but E just has to be unprovable in ENT. What this means, though, is that there are expressions in ENT which cannot be proven to be either true or false (in ENT). Thus ENT is incomplete. (This is the definition of incomplete).
The same line of proof applies to any sufficiently complex theory, including 'all of mathematics' (in a very loose sense), so it basically shows that there are true statements about maths that maths can't prove.