Core Idea - Connect Component Representatives

Pick one node from each component as a representative. For example, use the smallest-numbered node in each component. Connect the representatives in a chain: $(rep_1, rep_2)$, $(rep_2, rep_3)$,., $(rep_{k-1}, rep_k)$. This uses exactly $k-1$ edges. Any chain works.

The problem does not ask for minimum weight, minimum number of edges. A linear chain is the simplest solution that always works. You can find components with DFS, BFS, or DSU before connecting them.