Not necessarily business. Just doing something. Typically you would be going to different places, and/or doing different things. Perhaps business, perhaps not.

I don't think it's short for anything, it just refers to being out of the house and being busy doing something. It could be anything, shopping, going to a restaurant, just about anything not business, school or work related.

When I was young it was explained to me as "going out of your home" so you could "go about town". To be out and about is to be out of your home and going about your daily activities. People frequently use it when they are going to work, running errands, and doing things like that. Hope this helps!