To disengage means to “free or disentangle” yourself or some object from another person or object. No, it doesn’t mean breaking off your engagement to your beloved — that’s “chickening out.”
Does your mind wander when someone (your fiancé, for example) talks to you? Well, your mind can disengage as well, that is, stop paying attention. Disengage has quite a formal, intellectual ring to it, even if used about a situation as physical as two armies “disengaging from combat.” It’s a technical term, and faintly clinical; the kind of word frequently used by politicians and diplomats.
release from something that holds fast, connects, or entangles
free or remove obstruction from