Intuitively the idea behind the Dedeking cuts is the following. Assuming that you "know" what the real numbers are, then you can associate to each real number $a$ the set
$$( - \infty ,a ) \cap \mathbb Q=: C_a$$
Now, a set $C_a$ satisfies the four axioms of definition of a Dedekind cut, and moreover, each Dedekind cut is exactly one and only one of the sets $C_a$. Indeed, if $C$ is a Dedekind cut, and $a= \sup(C)$ then $C=C_a$.
Because of this, the correspondence $a \leftrightarrow C_a$ is a bijection between the real numbers and the set of Dedekind cuts.
Now, the issue is what to do if you don't know the real numbers, but you want to construct them. Well, you can construct the Dedekind cuts without assuming the existence of a real number. This way we can give an axiomatic construction for the reals.
And always remember, when we think about Dedekind cuts as "real numbers", what we mean is that, once we formally define $\mathbb R$, for each cut $C$ there exists an unique real number $a$ such that the cut $C$ is exactly
$$C= (- \infty, a) \cap \mathbb Q$$