(◊ ∃x Fx) ↔ (∃x ◊ Fx) can be seen as a conjunction of
(◊ ∃x Fx) → (∃x ◊ Fx) (the Barcan formula in the narrower sense)
(∃x ◊ Fx) → (◊ ∃x Fx) (the converse Barcan formula).
The forward direction, (◊ ∃x Fx) → (∃x ◊ Fx), says that no new objects come into existence when going from one possible world to another: If there is an accessible world where there exists an x s.t. Fx, then this x already exists in the current world (and Fx is possible at our world since we know it is true in the other world), so the object x that exists in that other world is not new. This property is called anti-monotonicity.
The converse direction, (∃x ◊ Fx) → (◊ ∃x Fx), says that no object ceases to exist when going from one possible world to another: If an x exists in the current world (and there is some accessible world where F is true of x), then there is an accessible world such that x exists in this world (and F is true of x in that world). This property is called monotonicity.
Together, (◊ ∃x Fx) ↔ (∃x ◊ Fx) expresses that the same set of objects exist in all possible worlds. It is hence an axiomatization of models with a constant domain, i.e. models where each world has the same set of individuals, whereas the combined Barcan formula is not valid in models with variying domains, where each world comes with a possibly different domain of objects.
If the model contains only one possible world, then the Barcan formula is trivially valid, since then we're talking about only one domain of objects anyway.