Link is good, but I guess direct explanation of this simple thing could be useful.

It is not hard to build explicit map between R and R² (more or less interleaving the binary notations for numbers).

So the claim of Continuum Hypothesis is:

For every property of real numbers P there exists such a property of pairs of real numbers, Q such that:

1) ∀x (P(x) -> ∃! y Q(x,y))

2) ∀x (¬P(x) -> ∀y¬Q(x,y))

(i.e. Q describes mapping from support of P to R)

3) ∀x1,x2,y: ((Q(x1,y)^Q(x2,y)) -> x1=x2)

(i.e. the map is an injection)

4) (∀y ∃x Q(x,y)) ∨ (∀x∀y (Q(x,y)-> y∈N))

(i.e. map is either surjection to R or injection to a subset of N)

These conditions say that every subset of R is either the size of R or no bigger than N.