In Theorem 2.40, Rudin talks about a $k$-cell $I$ formed by the intervals $[a_1, b_1], \ldots, [a_k, b_k]$. We split each interval at its midpoint $c_j = \frac{a_j + b_j}{2}$ and end up with $2k$ intervals $$ [a_1, c_1], [c_1, b_1], \ldots, [a_k, c_k], [c_k, b_k]. $$ He then says that the intervals $\{[a_j, c_j]\}$ and $\{[c_j, b_j]\}$ determine $2^k$ $k$-cells $Q_j$ whose union is the $k$-cell $I$.
Now, from an intuitive point of view, this is not hard to see. A line will be split into 2 regions, a square into 4 regions, a cube into eight regions, etc. For each dimension, we must choose either the interval $[a_j, c_j]$ or the interval $[c_j, b_j]$, and since there are $k$ binary choices, there are $2^k$ total of these cells.
But what does "determine" mean formally? Is it inconsequential for the rest of the proof? I tried showing it with induction, but without knowing what I'm really trying to show, a proof is fruitless.
I think you answered your own question. The intervals $[a_j, c_j]$ and $[c_j, b_j]$ for $1 \le j \le k$ determine $2^k$ $k$-cells, that is, there are $2^k$ $k$-cells
$$J_1 \times J_2 \times \dots\times J_k$$
where $J_i$ is either $[a_i, c_i]$ or $[c_i, b_i]$, such that their union is $I$.