simplex $B^{-1} \cdot A_j$ tableau interpretation In an iteration of the simplex tableau implementation, what is the interpretation of the columns $B^{-1} \cdot A_j$ underneath each variable $x_j$?

Dantzig & Thapa (Section 3.5) call it the "representation of the sth activity in terms of the basic set of activities" where in your case activity $s$ is $x_j$. In other words the vector describes the change of the values of the basic variables per unit of change of the non-basic variable $x_j$.
Example: solving
$$\begin{align}
\text{max.}\qquad & z=5x_1+6x_2,\\
\text{s.t.}\qquad & 2x_1+3x_2 \leq 18, & \text{(1)}\\
& 2x_1+x_2 \leq 12, & \text{(2)}\\
& x_1,x_2\geq 0,
\end{align}
$$
you get the tableau
$$\begin{array}{cccccc}
\hline
x_1  &   x_2  &   s_1   &  s_2  &   z  & \text{RHS}\\           
\hline
0    &  1    &  0.5   & -0.5 &  0   &   3\\      
1    &  0    &  -0.25 & 0.75 &  0   &   4.5\\
\hline    
0    &  0    &  1.75  & 0.75 &  1   &   40.5 \\
\hline  
\end{array}$$
with optimal solution $x_1=4.5$, $x_2=3$. Graphically, you can see the vectors of the non-basic variables $s_1$ and $s_2$ pointing into the direction of change:

