Theorem: If the matrix $\bar{A}$ is the result of applying a row operation $R$ to the matrix $A$, and if $E$ is the matrix that results from applying $R$ to the identity matrix $I$, then $\bar{A}=EA$
Proof:
The author provides a proof for which I am unclear and uncomfortable.
Suppose the $R$ is the operation of adding $c$ times row $s$ in $A$ onto row $r$ for $r \neq s$. The action of $R$ on the $n \times n$ identity matrix produces the elementary matrix $E$ with these entries:
$$E_{ij}=\delta_{ij}+c \delta_{ir} \delta_{sj}$$
I am unable to understand the existence of $\delta_{ir}$ and why the row r appears as a column index. Any help is appreciated
Because $\delta_{ir}=1$ if and only if $i=r$. You want to add something to the $i$th row only if it is the $r$th row.