Goldbach's conjecture is that "every even number bigger than 2 is the sum of exactly two prime numbers", so 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, etc.
For this statement to be *true* it just means that every even number there must exist two primes that add to that number. This is a statement about infinitely many integers.
A *proof* of Goldbach's conjecture consists of a finite number of formal reasoning steps that start with some axioms and end up at the statement of the result.
To this day, it seems as though Goldbach's conjecture is true. It holds for every number we've been able to test. However, no one has proved it is true or false yet (or proved that it is unprovable).
The proof of Gödel's result's involves very carefully formalizing what statements and proofs mean so that they can be encoded as statements about arithmetic. He then shows there is a statement with encoding G that says "The statement with encoding G cannot be proved" – if it is true, then it cannot be proved.
It's kind of confusing at first, but there is an easier way to get an intuition why there might be true statements that cannot be proved. Think of each statement about the natural numbers as a subset where each number in the subset makes the statement true. There are uncountably many subsets of the natural numbers (by Cantor's diagonalization argument). Proofs are finite chains of finite statements so there are only countably infinitely many of these. Therefore there must be subsets/statements that are true that do not have a matching proof.
The approach that Gödel's proof takes is not too different to the above argument – it is essentially a diagonalization argument – the complexity is in making the encoding of statements are numbers very precise.