I assume you're talking about words like "organize," "apologize," etc. It isn't an American innovation. Both "ize" and "ise" have been used in English for a long time — if the word was borrowed from French, it was generally spelled with "ise", while if it was adopted directly from Latin, it was spelled with "ize." The influence of Webster's dictionary and other such publications tended to standardize the spelling in US English to "ize," mostly (although not entirely, since Americans also write "advertise" and "compromise") while things remained more open in UK English, and indeed the OED allowed both and even listed "ize" first. In the 1990s, however, British editors switched to prescribing "ise." The linguist Lynne Murphy argues that when the British got online and started seeing Americans using "ize" all the time, they became convinced that this spelling was un-British. She quotes a writer during the 1990s who said that according to Fowler's Modern English Usage, either "ise" or "ize" is acceptable in the UK while "ize" is allowed in America, so he was only going to use "ise," because "speaking purely personally, I think that anything that is compulsory in America should be avoided."

So Americans use Z's because of nineteenth century dictionaries, while British people use S's because they're asserting their national identity.