Drag from one landmass to another (or tap a node, then tap the next). A bridge turns red once you’ve crossed it. You can’t reuse bridges.
Why it’s impossible (Euler’s 1736 result)
In graph terms, each landmass is a vertex and each bridge is an edge. A route that uses every edge exactly once is an Eulerian trail. Such a trail exists only if the graph has either 0 or 2 vertices of odd degree (odd number of incident edges).
In Königsberg there are four landmasses and all four have an odd degree, so no Eulerian trail exists. You’ll always get stuck with bridges remaining!