What does the old saw, "Shiver me timbers!" actually mean?

It's an old sailing expression expressing fright or surprise. It refers to the shaking of the ship's wooden frame (timbers) during a storm or battle.

Thank you both—alack that there is not a Best Answers (plural) citation. So it appears that “timbers” is the operative word in the phrase: the equivalent, whether or not with allusion to ship sailing of yore, of the contemporary notion of being “rattled.” As for shiver, I noted also its use in maritime argot to describe the unwanted flapping of sails when bringing ship’s course “too close to the wind”—probably a matter of concern but not to any such degree as spine shivering timber shivering. Surely seafarers of old must have been or become addicts of adrenaline rush.

Old ships were made of wood. If they were involved in a collision the timbers would 'shiver', ie vibrate, and there would be a possibility of the ship sinking. Also in heavy seas the struts throughout the ship would shake, shiver, thus alarming the sailors. The expression is a mild oath.