The listing of all the assets of a company in its Balance Sheet so that investors are clear about exactly what is involved in generating profits. Some types of transactions - contract hire, and rental - do not involve the company owning the item, but yet they still have use of it - and these may be 'Off Balance Sheet' and not identifiable as used in the business. 'On Balance Sheet' is a term to describe rules (see FRS 17) that state when assets must be listed.
Good old fashioned accounting rules! The original idea was that every asset used in a business must be listed in the Balance Sheet for every one to see and feel! Some contracts to use assets were drafted in such a way that they were rented, or available on Contract Hire - which created an excuse NOT to include them. This gave a false view that the profits originated from a lower Balance Sheet base than was actually the case. This activity as redefining as Contract Hire etc is called 'Off Balance Sheet'.