Why Back Edges Prevent Bridges

A back edge creates a cycle. Any edge in a cycle has an alternate path around the cycle. If tree edge $(u, v)$ has a back edge from the subtree of $v$ to an ancestor of $u$, then $(u, v)$ is not a bridge.

The back edge provides an alternate route. If no such back edge exists, $(u, v)$ is a bridge. Removing it disconnects the subtree of $v$ from the rest of the graph. The subtree has no escape route.