Is a matching $M$ maximum iff the graph doesn't have an augmenting path wrt $M$?
An augmenting path must alternate between gold and black, starting and ending with black edges. In this way, if you replace the gold edges with the black edges, you obtain a larger matching. This argument shows that a matching can only be maximum if there is no augmenting path, and Berge's theorem states the inverse direction: if a matching has no augmenting path then it is maximum.