There’s a widespread belief that these spellings belong only to American English, and that British English should use the ‘-ise’ forms instead, i.e. realise, finalise, and organise.
In fact, the ‘-ize’ forms have been in use in English spelling since the 15th century: they didn’t originate in American use, even though they are now standard in US English. The first example for the verb organize in the Oxford English Dictionary is from around 1425, from an English translation of a treatise on surgery written by the French physician Guy de Chauliac:
The brayne after ţe lengţ haţ 3 ventriclez, And euery uentricle haţ 3 parties & in euery partie is organized [L. organizatur] one vertue.
The OED’s earliest example for realize dates from 1611: it’s taken from a definition in A Dictionarie of the French and English Tongues, a bilingual dictionary written by Randle Cotgrave:
Realiser, to realize, to make of a reall condition, estate, or propertie; to make reall.
The first recorded use of the verb with an ‘-ise’ spelling in the OED is not until 1755 – over a century later!
The use of ‘-ize’ spellings is part of the house style at Oxford University Press. It reflects the style adopted in the first edition of the Oxford English Dictionary (which was published in parts from 1884 to 1928) and in the first editions of Hart’s Rules (1904) and the Authors’ and Printers’ Dictionary (1905). These early works chose the ‘-ize’ spellings as their preferred forms for etymological reasons: the -ize ending corresponds to the Greek verb endings -izo and –izein.