In the Hopcroft-Karp algorithm, what is the purpose of the breadth first search?
Vertex-disjoint means that no vertex appears in two augmenting paths.  The paths have to be disjoint so that all can be used.  If we have two augmenting paths, P1 and P2, and we augment the matching using using P1, is P2 still an augmenting path for the augmented matching?  Yes, if P1 and P2 are vertex disjoint.  

As for minimum length, it has the practical implication that we spend the shortest time looking for augmenting paths, but there is a technical matter, too.  Augmenting paths found this way are guaranteed to be vertex-disjoint.  Look at this text.