Problem to understand a Bellman Ford algorithm exercise
The only way to understand it is to break it down.

Given a weighted, directed graph $G=(V, E)$ with no negative-weight
cycles, let $m$ be the maximum over all vertices $v \in V$ of the
minimum number of edges in a shortest path from source $s$ to
$v$. (Here, the shortest path is by weight, not the number of edges.)

Okay, so $s$ is fixed, and let us suppose you have a vertex $v$.
Now we look at all the cheapest paths from $s$ to $v$ and we take the
path that has the fewest edges; this number of edges we refer to as $m_v$.
If we repeat this for every vertex $v \in V$, then we get one $m_v$ for
each $v$.  The value $m$ is the maximum of all $m_v$s, or $\max_{v \neq s}m_v$.
Or in other words, over all cheapest paths starting in $s$, let us look at
the vertex with the longest shortest.  It has length $m$.
It is confusing because we want the longest shortest over all the cheapest.