Understanding the Leontief inverse

The equation you are concerned with relates total output $x$ to intermediate output $Ax$ plus final output $d$, $$ x = Ax + d $$.

If the inverse $(I - A)^{-1}$ exists, then a unique solution to the equation above exists. Note that some changes of $a_{ij}$ may cause a determinate system to become indeterminate, meaning there can be many feasible production plans.

Also, increasing $a_{ij}$ is equivalent to increasing the demand by sector $i$ for the good produced by sector $j$. Thus, as sector $i$ produces more, it will consume more of sector $j$'s goods in its production process.

The formulation $(I−A)^{−1} =I+A+A^2 +A^3 \cdots$ is the most interesting one. $I*d$ is the production of d itself, $A*d$ is supply of intermediate goods and services to the direct producers of d, $A^2d$ is supply of intermediate g&s to these, and so on and so on.