Godel's self-reference lemma states: Let $\psi(v_{1})$ be an $L_{NT}$ (language of num theory) formula with only $v_{1}$ free. Then there is a sentence $\phi$ such that $N \vdash (\phi \iff \psi([\phi]))$
, where $[\phi]$ is $\phi's$ Godel number.
Apparently here $\phi$ is saying of itself "$\psi$ is true of me". But I don't understand precisely how and am hoping someone can explain (I must be missing something with the Godel numbers).
The formula $\psi(x)$ is an arithmetical formula: it makes a claim about numbers. However, the Godel numbering is done in a way that every logic expression gets a unique number. Therefore, by making a claim about a number, the claim can be seen as making a claim about the logic expression that that number is the Godel number of. (Note: depending on how the numbering is done, not all numbers are the gode number of any expresion at all, but you will never have two different logic expressions with the same Godel number)
Godel's self-reference Lemma says that for any formula $\psi$ that makes a claim about a single number, there is a sentence $\phi$ that is equivalent to the sentence $\psi([\phi])$ which, since it is a sentence about the godel number $[\phi]$ of expression $\phi$, can be seen as a claim about $\phi$ itself.