Biz is just onomatopoeic for the first syllable of business.

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