Histogram - Why It Works

When you pop bar $j$ because bar $i$ is shorter:

$1.$ Bar $j$ can't extend right past index $i - 1$ (bar $i$ is shorter).

$2.$ Bar $j$ can't extend left past the new stack top (that bar was shorter than $j$).

$3.$ Between those boundaries, all bars are at least as tall as $j$ (they were pushed after $j$ and before being popped). So the width calculation $(right - left - 1)$ is correct.