The lemma as stated in Enderton's logic says:
Fixed-Point Lemma. Given any formula $\beta$ in which only $v_1$ occurs free, we can find a sentence $\sigma$ such that
$$ A_E \vdash [\sigma \leftrightarrow \beta(\mathbf S^{\sharp\sigma}\mathbf 0)]. $$
We can think of $\sigma$ as indirectly saying, “$\beta$ is true of me.”
I wonder what does "$\beta$ is true of me" mean?
Can the lemma be interpreted as:
For any formula $\beta$ having one free variable $x$, there is a sentence $\sigma$ which we can think of $\beta$ as saying $\sigma$ is true.
Is that what the lemma says indirectly? If not, What is the right interpretation of it then?
First of all, we are considering formulae (like $\beta, \sigma$) in first-order language of arithmetic; thus, their "standard" interpretation are assertions about numbers.
We can think of $\beta(v_1)$ with $v_1$ free, as a sort of "function" (a propositional function) that maps numbers into assertions about them; i.e. $\beta(\mathbf S\mathbf 0)$ is an assertion (true or false) about the number $1$, ans so on with $\beta(\mathbf S\mathbf S\mathbf 0)$, etc.
The Fixed-Point Lemma asserts that, for any such $\beta$ we can find a sentence $\sigma$ (in the first-order language of arithmetic) such that the "function" corresponding to $\beta$ maps the Gödel number of $\sigma$, i.e. the number $\sharp\sigma$, into the "assertion" $\sigma$ itself.