You’re most likely to come across this moderately rare word in either an heraldic context or a scientific one. In heraldry, it refers to a element of the design that has a narrow border around it. For example, the flag of the UK is described in the 1801 Act of Union, in the extraordinary language peculiar to the heraldic craft, as
Azure, The Crosses Saltire of St Andrew and St Patrick, Quarterly per saltire, countercharged Argent and Gules, the latter fimbriated of the second, surmounted by the Cross of St George of the third, fimbriated as the Saltire.
The fimbriated bits are those narrow borders of white (argent) whose positioning is so important in ensuring that the flag isn’t being flown the wrong way up.
When used as a scientific description, it refers to some part of an animal or plant that has a fringe of hairs or the like. It is also sometimes used as an elevated term to describe something that’s hairy or has a fringe:
One old fellow, who looked like a pirate with his red-rimmed eyes, weather-beaten skin, and fimbriated face, grinned up at me in such sardonic challenge that I walked directly in front of him and began to speak.
The Story of a Pioneer, by Anna Howard Shaw, 1915.