On the chess board few squares are marked. Show that the minimal number of ranks and files which cover all marked squares is the same as the maximal number of rooks on the marked squares which do not threaten each other.
I think it can be re-represent with the concept of vertex cover and graph matching. Here is the definition:
but I am very confused about how to re-represent this question in terms of vertex covers concepts. For example, in the chess question ,what should be treated as vertices and what should be treated as edges.
Since we need a bigraph to apply the theorem, let's suppose that the bigraph goes from $\{1,...,8\}$ to $\{1,...,8\}$ (it could be any number, but since you wrote chess board, I've taken an $8 \times 8$ board), with an edge $(i,j)$ if the box on row $i$ and column $j$ is marked.
Now, the minimum number of ranks and files covering the marked squares, is the minimum vertex cover, since any edge corresponds to a marked square, and the ends of the edge correspond to the rank and file of the respective marked square.
Similarly, a maximal matching is a set of non-intersecting edges. Each edge corresponds to a square, so taking a collection of edges is like taking a collection of squares. If edges don't intersect, then the marked squares on the chessboard neither have the same row nor the same column, therefore if you place rooks on these squares, they won't attack each other. Therefore, the maximal number of rooks with which this is possible, is clearly the size of the maximal matching.
Now the theorem gives you the desired result.