"To the bitter end" is a phrase that means "to the last degree," "until all things have been tried," "to the limits of one's efforts," or "to the last extremity."
However, it is not certain if the phrase "the bitter end," was known before Captain Smith's writing or not.
The nautical connection: The anchor is shackled to the anchor cable (or anchor chain), the cable passes up through the hawsepipe, through the pawl, over the windlass gypsy (or wildcat) down through the "spurling pipe" to the chain/cable locker under the forecastle (or poop if at the stern (US fantail)) - the anchor bitts are on a bulkhead in the cable locker and the bitter end of the cable is connected to the bitts using the bitter pin, which should be able to be released from outside the locker to "slip" the anchor. This would occur if the windlass brake has slipped in a storm for example and you have reached "the bitter end". It originally applied in sailing vessels where the cable was a rope, and the windlass or capstan was powered by many sailors below decks.