If $G$ has a perfect matching, it has an even number of vertices. Suppose there's an edge $e$ whose removal disconnects $G$. Removing that edge splits $G$ into two pieces. Either each piece has an even number of vertices, or each piece has an odd number of vertices.
If each piece has en even number of vertices, then $e$ can't be part of a matching, since the other edges in the matching must each be wholly in one of the two pieces, so you must get a matching on each of the pieces, but once you've used $e$ each piece has an odd number of veertices.
If each piece has an odd number of vertices, then $e$ must be a part of every matching, and you can't use any edge that meets $e$ at a vertex. For if you use such an edge, that leaves an odd number of vertices in the piece containing that edge, and there's no matching there.