to get safely out of danger, or to get someone else safely out: to escape, or to rescue
At its heart this word is about slipping free of a threat that is closing in. Picture someone wriggling out of a trap, darting away from a sword, or being pulled clear of a flood just in time. The danger is real and present, and the action is the narrow getaway.
The word turns naturally between escaping yourself and getting someone else out. When a person is the one slipping free, it means to escape or get away (David escapes Saul, Lot is told to flee for his life). When the action falls on someone or something else, it means to rescue, deliver, or save (to save a life, to carry valuables to safety, to spare a city). It is the same motion of slipping clear of harm, only seen from the side of the one doing the saving.
Because the rescue is always from a specific, pressing danger, the word carries a note of urgency and narrow margins. Often it describes barely making it out, the lone survivor of a slaughter, the bird breaking free of the snare just as it snaps shut.