A recursive solution to the all-pairs shortest-paths problem
(1) tells you how to compute the shortest path. 

Note that given that the graph has no negative-weight edge, the shortest path between two nodes is a simple path (with no cycles) which has at most $n-1$ edges. More than $n-1$ edges means a cycle which increases the path length. But we are interested in the shortest path.  

(2) claims that after $n-1$ edges the shortest path length remains same:
$L_{i,j}^{n-1}=L_{i,j}^{n}$ is read as "the shortest path from $i$ to $j$ passing through $n-1$ edges is the same as the shortest path through $n$ edges".

In fact it means that there is no need to compute $L_{i,j}^{n},L_{i,j}^{n+1},\dots$ since it will not lower the path between $i$ and $j$.