What does it mean that the continuum hypothesis is independent of ZFC?

I know very roughly that ZFC is a system of axioms of set theory and the continuum hypothesis states that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. It says in Wikipedia that you may add that proposition or its negation to the axioms of ZFC and the resultant system will be consistent iff ZFC is consistent. And I think consistent means impossible to derive a contradiction.

I don’t understand the significance of this result, though. Does it roughly mean that the continuum hypothesis is completely impossible to answer, or that it’s both true and false, or definitely false, or something else? I don’t think it seems to be definitely true, whatever is happening.