sever, cut off, interrupt
斷 centers on the idea of forcefully separating or stopping something: severing, cutting off, or interrupting a connection or flow.
斷 is the traditional form of 断. It combines 㡭 (a component related to cutting or severing) with 斤 (an axe), together expressing the idea of cutting or severing with a tool. The modern simplified form 断 replaces the left side with a simpler shape.
The left side 㡭 looks like tangled threads or connections, and the right side 斤 is an axe. Imagine using an axe to sever the tangled mess, cutting it off completely.
For ダン, picture a dam (ダン) being built to cut off a river's flow, just as 斷 severs or interrupts.