How to construct an ordinary matching from a fractional matching?
It is not possible to get a maximum matching from any fractional matching by only looking at the values of the fractional matching. Consider the undirected triangle graph with  vertices $v_1,v_2,v_3$ and edges $e_1=(v_1,v_2), e_2=(v_1,v_3), e_3=(v_2,v_3)$. If we assign $f(e_1)=f(e_2)=f(e_3)=\frac{1}{2}$, this is a valid fractional matching. However, an actual matching can only contain one of the edges, so we cannot distinguish these edges by their value.

It is of course always possible to obtain a matching from a fractional matching by first selecting all edges with at least a certain fraction and then removing edges such that you end up with a matching. This matching, however, is usually not a maximum matching.