Section 5 (after (5.4)) — f_i^(k) is the minimum time over paths with at most k stops

Let $N = n + 1 \ge 2$ and $t_{ij} > 0$ for $i \ne j$, and let $f^{(k)}$ be the successive approximations (5.1) from the direct-route policy (5.2), with $f_N^{(0)} = 0$. Then for every $k \ge 0$ and every city $i$, $f_i^{(k)}$ is the minimal time to travel from $i$ to $N$ along a route with at most $k$ stops. That is, some route from $i$ to $N$ with at most $k + 1$ roads has time $f_i^{(k)}$, and every such route has time at least $f_i^{(k)}$:

The page states this for $k = 1$ ("$f_i^{(1)}$ represents the minimum time for a path with at most one stop"). For general $k$ it invokes the same fact as "the physical interpretation of this iterative scheme", on which both (5.5) and the $N - 1$ bound rest.

Formalization Note At $i = N$ the trivial route (no road, time $0$) is admitted, matching $f_N^{(k)} = 0$. Routes may repeat cities; with positive times this does not change the minimum. The convention $f_N^{(0)} = 0$ is the corrected reading of (5.2) explained in the (5.4) item.