From a memorable, intuitive perspective, your account is fine. The considerations of degrees of freedom rest on the understanding that each standardized residual,
$$Z_i = \frac{O_i-E_i}{\sqrt{E_i}},$$
is close enough to having a standard Normal distribution that the sum of their squares
$$X^2 = \sum Z_i^2 = \sum_i^n \frac{(O_i-E_i)^2}{E_i}$$
has a distribution close to that of the sum of $n$ standard Normal variables. (For convenience I have written $n=rc$ and index the residuals from $i=1$ to $i=rc$, rather than following the double-indexing that is usually used in two-way tables.)
The problem, as every textbook at every level points out, is that the $Z_i$ are not independent. In fact, they satisfy a lot of linear relations. Across each row, for instance, the sum of the $O_i$ (counts of observations) equals the sum of the $E_i$ (their expectations, constructed to give the same sum), whence the sum of the $Z_i$ across each row is zero. Similar, the column sums of the $Z_i$ are zero. But many more sums than that are zero: any linear combination of such sums will also be zero.
The best solution is to consider this geometrically. The vector $\mathbf{Z}=(Z_1, \ldots, Z_n)$ can be located anywhere in $\mathbb{R}^n$. (Visualize this in $n=3$ dimensions.) If the $Z_i$ truly were independent, they would have a rotationally-invariant distribution around the origin. The only variation in their probability density would be radial. Indeed, $X^2$ is precisely the squared length of $\mathbf{Z}$.
Any single sum-to-zero constraint defines a hyperplane in this space. (Visualize a plane in $\mathbb{R}^3$, such as the $xy$ plane.) The constraint restricts the vector $\mathbf{Z}$ to lie within that hyperplane. Nevertheless, because any rotation in that hyperplane can be extended to a rotation of the entire space--just fix the perpendicular direction--the distribution of $\mathbf{Z}$ remains rotationally invariant in this hyperplane. It therefore is identical to the distribution of independent standard Normal variables within the hyperplane itself, which has one less dimension, $n-1$.
(In the running example, $\mathbf{Z}=(Z_1,Z_2,Z_3)$ when restricted to the $xy$ plane in $\mathbb{R}^3$ is just $(Z_1,Z_2)$. Therefore its squared length is just $Z_1^2 + Z_2^2$, which by definition has a $\chi^2(2)$ distribution. Due to the rotational invariance of $\mathbf{Z}$, the distribution of $|\mathbf{Z}|^2$ in any plane through the origin in $\mathbb{R}^3$ will be the same as this one: namely, $\chi^2(2)$, not $\chi^2(3)$.)
In this fashion each additional hyperplane thereby reduces the number of independent standard Normal variables in the description, provided it truly reduces the dimension of the space in which $\mathbf{Z}$ may lie. The actual mathematical content of the $\nu = (r-1)(c-1)$ equation is to assert that (a) the $r+c$ obvious sum-to-zero constraints in the $r\times c$ table amount only to $r+c-1$ independent constraints--there is one degree of redundancy--and (b) there are no other constraints independent of them. Proving these facts is a matter of linear algebra: that is, of solving systems of equations and counting dimensions.
(It is easy to see there are at most $r+c-1$ independent constraints, because the fact that all $n$ $Z_i$ sum to zero follows by adding either all the row sums or all the column sums, showing there is at least this degree of redundancy among the row-sum and column-sum constraints.)
Consequently, we should visualize the situation as one of looking at the distribution of $rc$ independent standard Normal variables within a linear subspace that is determined by the intersection of $r + c - 1 $ independent hyperplanes. Within that subspace, the sum of squares of those variables still gives the distance to the origin. That distance, however, lies within a space of dimension only
$$\nu = rc - (r + c - 1) = (r-1)(c-1).$$
Thus, the distribution of the sum of squares is the same as the distribution of the sum of squares of $(r-1)(c-1)$ independent standard Normal variables.
The answers at How to understand degrees of freedom? provide additional explanations, many from similar viewpoints but at varying levels of sophistication and rigor.
