While defining area axiomatically, Apostol makes the statement following statement:
The term "rectangle" as used here refers to any set congruent to a set of the form $$\{ (x,y)|0\le x\le h\ ,\ 0\le y\le k \}$$
with congruence defined in the footnotes as follows:
Two sets are said to be congruent if their points can be put in one-to-one correspondence in such a way that distances are preserved. That is, if two points $p$ and $q$ in one set correspond to $p'$ and $q'$ in the other, the distance from $p$ to $q$ must be equal to the distance from $p'$ to $q'$; this must be true for all choices of $p$ and $q$.
What exactly does he mean by "correspond" here? Obviously, I have an intuitive idea of what he's trying to get at here but I can't figure out what a pair of points in the two sets corresponding with each other means precisely.
The correspondence is usually called an isometry (or congruence). It's a one-to-one and onto transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ (a bijection from the plane to itself). This means that the transformation has an inverse as well, which you can show is also an isometry.
The correspondence is between a point $(x_0, y_0)$ and its image under this transformation, say $$ (u_0, v_0) = T(x_0, y_0). $$
The distance preserving condition can be written as $$ d\bigl( (u_0, v_0),\, (u_1, v_1) \bigr) = d\bigl( (x_0, y_0),\, (x_1, y_1) \bigr) $$
These isometric transformations fall into a few familiar families:
which you can read much more about here.