Walkthrough

Say $n = 5$, tree: $1 \to 2 \to 3 \to 4 \to 5$ (a chain). Query: $(5, 3)$ (find $3$rd boss of employee $5$). Build $up$: up[5][0] = 4, up[5][1] = 3, up[5][2] = 1. Now jump: $k = 3 = 11_2$ (bits $0$, $1$ set). Jump $2^1 = 2$: v = up[5][1] = 3. Jump $2^0 = 1$: v = up[3][0] = 2. Answer: $2$.

You reached the answer in $2$ jumps instead of $3$ single steps. For large trees, the savings are dramatic.