Biz is just onomatopoeic (a way of representing the sound in writing) for the first syllable of business. If you shorten the word business without doing this, it would be bus; most speakers would pronounce this as if it were a large vehicle for transporting passengers.
The most famous usage is "show biz" (often written as "showbiz"). This is merely a glitzy way of saying "show business." The image conjured by this usage is that of a 1930s director using it as a fancy bit of jargon to impress his friends.
The more modern usage of biz is as the end of a URL: e.g. madeupcompany.biz