English idiom. A place for everything, and everything in its place.
Quite simply, everything is hunky-dory, all in order, neat, tidy and in good trim.
Sailing ships were relatively small places, and complex, to boot. Consider the sheets (that's ropes, to we landlubbers), sails, block and tackle and rigging (more ropes). Think of the crew, scurrying about their various duties in all weathers on a slippery and pitching deck. If ropes were not properly coiled and stowed, rigging tweaked and everything in generally good order, a dangerous state of chaos would reign onboard. Shipshape, therefore simply means that everything is neat and tidy.
The "Bristol" bit is quite interesting. Anyone who knows Bristol well is aware that the tidal range is enormous - up to 35 feet (10 meters). Ships tied up in the harbour were subject to this massive rise and fall, and any vessel which was not well-built, well secured and with a well-distributed and well-stowed load was likely to suffer damage.
There is, however, a cloud on the horizon. Some nitwit decided that it was now politically incorrect to use this phrase as it was somehow connected with the slave trade (ships sailing from Bristol often being involved in the slave triangle running between England, Africa and the West Indies). Despite the fact that slaves were rarely taken to or from Bristol, it's now deemed, in some circles, to be an inappropriate phrase, as "niggardly" was once, briefly.