This is the 10303rd most frequent Russian word.
британскими
"Британскими" means "British" (instrumental or plural form in Russian).
In this sentence, 'британскими' functions as a plural instrumental adjective describing the noun 'писателями' (writers), indicating nationality.
Эти книги были написаны британскими писателями.
These books were written by British writers.
Here, 'британскими' serves as a plural instrumental adjective, modifying 'традициями и обычаями' (traditions and customs), indicating the origin or style.
Мы увлеклись британскими традициями и обычаями.
We have become interested in British traditions and customs.
'Британскими' describes 'художниками' (artists) in the instrumental plural case, specifying their origin.