The theory of mammals includes several axioms:
- Female mammals produce milk
- Mammals are warm-blooded
- Mammals have hair
There are many creatures that satisfy these axioms and are therefore mammals. These include the giraffe, the elephant, and the tree shrew. We can call these models for the theory of mammals.
We can ask questions like Do all mammals have hair?. Answer: yes, because this follows (trivially) from the axioms. There might be other questions where the answer is yes but not quite so trivially. For example: Are all female mammals warm-blooded? Answer: yes, because every female mammal is a mammal. Et cetera.
There are other questions for which the answer is certainly "No", like Are there any cold-blooded mammals?
Those are questions for which the answer follows logically from the axioms. But there are other kinds of questions we could ask, like: Do mammals have trunks? Answer: There are some mammals (i.e. models of the theory of mammals) that have trunks and others that don't.
ZFC is a theory about universes of sets --- it is a list of axioms, like "There exists an infinite set". There are many different universes of sets, each of which is called a model of ZFC if it satisfies those axioms.
We can ask questions about those universes, like Does every universe contain an infinite set?. Answer, yes, because that follows from the axioms.
We can also ask questions regarding which the axioms are silent, like Is there a set with cardinality greater than the natural numbers but smaller than the set of subsets of natural numbers? (This is a form of the continuum hypothesis.) It is not immediately obvious whether the axioms dictate the answer to this question. But Kurt Godel gave an example of a universe of sets (that is, a model of ZFC, where all the axioms are satisfied) where the answer is "no", whereas Paul Cohen gave an example in which the answer is "yes". Those are two models among a vast array of others.
(This is just like proving that the theory of mammals can't decide whether all mammals have trunks by pointing first to an elephant and then to a zebra.)
Bottom line: Just because you have a bunch of axioms, it does not follow that those axioms can tell you everything about all the models of those axioms. We have axioms for mammals, and for universes of sets, but those axioms can't tell us whether a particular mammal has a trunk, or whether a particular universe of sets satisfies the continuum hypothesis.
PS---I think (but am not sure) that some of your confusion might arise from thinking of ZFC as a set of axioms for sets, whereas it is better to think of them as axioms for universes of sets.