Consider the following information:
In set theory, the axiom scheme of separation aka specification states that, given any set $X$ and any property $P$ of the elements of $X$, there is a set $$\{X \mid P\} = \{a \in X \mid P(a)\}$$ consisting precisely of those elements of $X$ for which $P$ holds: $$ a \in \{X \mid P\} \leftrightarrow a \in X \land P(a).$$
What is the formal and rigorous definition of "$P$ holds" and of "$a $ property of the elements of $X$"?
In set theory, typically there is only one type of object (sets) and properties are first-order formulas in the language of set theory (the language with only the relation symbol $\in$, as well as the logical relation symbol $=$). So the idea that a property is specifically a property "of the elements of $X$" doesn't really have any formal content. Formally, any property applies globally to every object (i.e. every set).
Now, informally, when we develop set theory, we extend the language to include numerous defined constants, functions, and relations$^*$, and it may well be the case that we consider properties that only really make sense on certain sets. For instance, we can write down a complicated formula in the language of set theory that means "$x\le 10$", which is a property that is intended only to apply to natural numbers, not to all sets. Still, our formula must "mean something" globally but in practice, it doesn't matter how we define it outside the natural numbers, there's just an informal requirement that we don't ever ask whether it holds on objects that aren't natural numbers.$^{**}$
But anyway, that digression is just intended to say that we often informally do think about whether a property is "of" a certain restricted set, even though formally they are all global properties.
As for what it means formally to say a property $\varphi$ holds at some element $x\in X$, it just means the same thing as to assert $\varphi(x).$ "Holds" is just a verb we use to match how we think about things informally. It will probably be helpful to understand what an instance of the axiom schema of separation actually looks like in the formal language of set theory. If we have some formula $\varphi(x,\vec y)$, the corresponding instance is$^{***}$ $$ \forall \vec p \forall X\exists b\forall z(z\in b \leftrightarrow z\in X\land \varphi(x,\vec p)).$$ Formally there is no verb "holds" here... the formal syntax of the formula $\varphi$ is just copied and pasted into the schema to give the resulting axiom.
$^*$ The definitional extension may be thought of as a formal process as well, but this doesn't much bearing on the point here as the relations and functions we define are global.
$^{**}$ This is sort of inevitable when we only have one type, and it is sometimes considered an unappealing aspect of set theory and the "malformed" statements we can prove as a result are called "junk theorems", though it's contentious whether this is actually all that bad, and to what extent other formalisms truly avoid "junk".
$^{***}$ Don't worry too much about the parameters $\vec p$... they are necessary since a given property might be expressed in terms of some variable sets but for the sake of understanding the concept here, you might as well assume they aren't there and $\varphi$ is just a single-variable formula.
My interpretation may be more syntactic than semantic.
A property (such as a unary predicate $P(a)$) is syntactically a formula. Formulas that have a proof are provable. In my view, a property holds if it has a proof.
So in order to prove $a \in \{ X | P\}$, you would have to prove that $a \in X$ and also that $P(a)$. This would let you prove the conjunction $a \in X \wedge P(a)$ which is the equivalent form of $a \in \{X | P\}$.
An example: Let the predicate Prime$(n)$ denote prime numbers. To prove that $7 \in \{\mathbb{N} | \text{Prime}\}$, you have to prove that $7 \in \mathbb{N}$ and that $7$ is prime (aka Prime$(7)$).
Remark: properties with free variables may have proofs also, it is not needed to put quantifiers at the front (such as $\forall x P(x)$), just the meaning is slightly different in some edge cases.
