An invariant is a property $P$ that is shared by all isomorphic graphs. In other words, a property $P$ is an invariant provided that whenever $G_1$ and $G_2$ are isomorphic graphs, if $G_1$ satisfies $P$, then $G_2$ also satisfies $P$.
A bridge is an edge $e$ whose removal yields a disconnected graph.
Illustration: Below I generated a random connected graph (the blue graph) and highlighted it's bridges in orange. Then I relabeled the vertices of the graph to form the red graph below highlighted it's bridges in orange too.
It's quite clear in the above example that bridges map to bridges and non-bridges map to non-bridges. The graph's structure has not changed. This is all isomorphism is, relabelling the vertices of the graph.
To prove that the existence of a bridge is a graph invariant formally, we should:
Let $G=(V,E)$ be an arbitrary graph with a bridge $ij \in E$. Let $\varphi$ denote a bijection with domain $V$. Show that $\varphi(i)\varphi(j)$ is a bridge in $\varphi(G)$.
Repeat the above for when $ij$ is a non-bridge.
Note: this proves the stronger (but unsurprising) result that the number of bridges is preserved under isomorphism.