Essentially, it is because e is on a simple circuit you can get to the vertices connected to e going the other way along the circuit.

Removing an edge will only ever remove the connectivity of a graph if that edge is the only way to get from one section of the graph to another - if that edge forms a bridge. As it is on a simple circuit this is not the case.

Here, we want to find loops. What can you say about the minimum number of edges of the graph? Can you use this result to show that a loop exists? That is to say, do we arrive at the same point at least twice?