"The score!" he burst out. "Three goes o' rum! Why, shiver my timbers, if I hadn't forgotten my score!"
Robert Louis Stevenson, Treasure Island, Ch. 8.
Wiktionary suggests this expression refers to a wooden ship being pounded by rough seas, so its 'timbers shake' and its sailors experience 'fear' and 'awe', so the sense is of shivering or trembling with fear, both ship and man.
But I don't think this explanation is correct. I think in this context shiver literally means split apart, the image is of the ship breaking up. One definition of shiver in the OED is 'To break or split into small fragments or splinters', recorded from the 12th century.
It's sometimes suggested that it's a purely literary or comic phrase but Etymonline gives several examples of very similar phrases in which 'my timbers' get started, smited or split, and suggests the tantalising idea that sailors might have used timbers as slang for arms and legs, so it could have been a profanity, similar to blimey, 'blind me'.
And what is the etymology of shiver? It's thought to come ultimately a Proto-Indo-European root *skei-, 'split', so it shares it origin with a fascinating medley of words, including
shin = a bit cut off
science = originally to separate things out, to know
nice = originally careless or ignorant, from Latin nescius, 'not knowing'
schism = a rift between factions
shit = waste got rid of
shingle = wooden rooftile
shed = cast off
ski = a piece of wood split off
To finish off here's a joke
'I' before 'E', except after 'C'. Disproved by Science.