There is a theorem that says the angles of a triangle always add up to 180 degrees. There is another theorem that says the angles of a triangle always add up to more than 180 degrees. How can there be two theorems that lead to two different outcomes? Easy: they are based on different situations. The first one is predicated on the axioms of Euclidean geometry. The second is based on the axioms of spherical geometry; every triangle (suitably defined) on a sphere has angles adding up to more than 180 degrees.
There's even a third system, where every triangle has angles adding up to less than 180 degrees; it's called hyperbolic (or, Lobachevskian) geometry.
Which one is the true geometry? Unask the question. There is no such thing as the true geometry, there are just different models that are useful in different circumstances.
The situation with the Continuum Hypothesis is similar. There is a set theory in which it is true, and a set theory in which it is false, and no one has come up with a convincing argument that one of these is the true set theory and the other a fake. For the time being, and perhaps forever, we have to accept that there are different models that are useful in different situations.
If it's any comfort to you, none of this has any effect whatsoever on the mathematics of the typical introductory discrete math class. Everything you are likely to see in such a class will hold regardless of what goes on with the Continuum Hypothesis.