How does the axiom of separation form a new set from an existing set?

The axiom schema of separation allows for the creation of a new set by selecting specific elements from an existing set that satisfy a given condition.1

Unlike the unrestricted principle of comprehension—which historically led to Russell’s paradox by allowing the formation of sets from any arbitrary condition—the axiom of separation imposes two critical restrictions to ensure consistency:12

• Preexisting Sets: It cannot be used to assert the existence of sets unconditionally. Instead, it must be applied to a set that has already been established.1
• Definite Formulas: It requires the use of a "definite" formula (a well-formed formula of ZFC) to define the condition $S(x)$ that elements must satisfy.1

Essentially, given an existing set $A$ and a condition $S(x)$, the axiom guarantees the existence of a new set containing precisely those elements $x$ within $A$ for which $S(x)$ is true. By "separating off" these elements from a previously defined set, the axiom avoids the logical inconsistencies associated with forming overly large or paradoxical collections, such as a universal set.12