In order to prove the continuum hypothesis is independent from the axioms of $ZFC$ what Cohen did was to start with $ZFC+V=L$ (in which the generalized continuum hypothesis holds), and create a new model in which $ZFC$ in which the continuum hypothesis fails.
First we need to understand how to add one real number to the universe, then we can add $\aleph_2$ of them at once. If we are lucky enough then $\aleph_1$ of our original model did not become countable after this addition, and then we have that there are $\aleph_2$ new real numbers, and therefore CH fails.
To add one real number Cohen invented forcing. In this process we "approximate" a new set of natural numbers by finite parts. Some mysterious create known as a "generic filter" then creates a new subset, so if we adjoin the generic filter to the model we can show that there is a new subset of the natural numbers, which is the same thing as saying we add a real number.
We can now use the partial order which adds $\aleph_2$ real numbers at once. This partial order has some good properties which ensure that the ordinals which were initial ordinals (i.e. cardinals) are preserved, and so we have that CH is false in this extension.
(I am really trying to avoid a technical answer here, and if you wish to get the details you will have to sit through some book and learn about forcing. I wrote more about the details in A question regarding the Continuum Hypothesis (Revised))