How does the Kronecker delta select coodinates? I'm reading A Gentle Introduction to Tensors. In introducing the Kronecker delta it says it "[...] it may be used as a coordinate selector, in the following sense:"
$\delta_j^i x^j = x^i$
Where superscript indices are rows, and subscript indices are columns. My current understanding is that this does nothing to $x$ besides change $i$ to $j$, which doesn't seem to be meaningful in this context, as it is still the same column vector. How does the Kronecker delta select coordinates? Is this just a tool to change tensor indices in equations?

Here is the intuition. Recall that repeated indices imply summation. To put an example consider 3 dimensions only. Hence
$$
\delta^i_jx^j = \delta^i_1x^1+\delta^i_2x^2+\delta^i_3x^3
$$
However, remember that $\delta_j^i=0$ for $i\neq j$ and $\delta_i^i=1$. Therefore, among $\delta^i_1,\delta^i_2,\delta^i_3$ only one of those 3 quantities is nonzero. For example if $i=2$ then $\delta^i_1=0,\delta^i_2=1,\delta^i_3=0$
Therefore, $\delta^2_jx^j=\delta^2_1x^1+\delta^2_2x^2+\delta^2_3x^3 = x^2$ and this is how the delta selects coordinates.
So in general among all terms involved in $\delta^i_jx^j$ just the $i$-th terms survives: the one with $x^i$. Therefore $\delta^i_jx^j=x^i$.
Moreover $\delta^i_jx^j$ is an operation over the whole vector $x$, and the result $x^j$ is just a component. This technique is one of the most useful rules to "play" with indices. Here is an example where this identity is useful apart from just "interchanging" indices in another answer of mine here.
