Trying to understand CLRS bucket sort analysis
You are asking why

$\qquad\displaystyle \sum_{j=1}^{n}\sum_{k=1}^{n}X_{ij}X_{ik} 
\quad=\quad \sum_{j=1}^{n}X_{ij}^2 + \sum_{j = 1}^{n}\sum_{1 \leq k \leq n,\\ k \neq j}X_{ij}X_{ik}$

holds (in expectation, but that's irrelevant for this; it might not always be, though!).

Note that the left-hand side and the second summand on the right-hand site are identical except for the $k \neq j$ constraint. That is, the difference between the two terms is exactly

$\qquad\displaystyle \sum_{j=1}^n \sum_{1 \leq k \leq n,\\k=j} X_{ij}X_{ik} \quad=\quad \sum_{j=1}^n X_{ij}X_{ij} \quad=\quad \sum_{j=1}^{n}X_{ij}^2$.

No deep insight here, just basic manipulation of sums.