Lowenheim Skolem theorem and Set theory

Question

I have read this sentence "If ZFC is consistent, then there is a countable model of it by Lowenheim Skolem theorem" in many text books of mathematical logic and set theory. The theorem tells us a word "countable", or " infinity cardinal". And this theorem is proved by axioms of ZFC. Here is my questions) Athough the Lowenheim Skolem theorem is proved in ZFC, how can we apply this to the set theory(ZFC)? Does this theorem should be applied to subtheory of ZFC? Is the theorem proved only in ZFC? Thank you so much for your answer.