It simply means the ordinary man on the street. It was introduced as an example in a trial in Britain in 1933 - although the phrase is older. Presumably though only dating back to the introduction of the Clapham bus service!
It does now have a definite legal meaning. You might for example have a contract with a supplier which is allowed to have more detailed and complex terms than one with a regular member of the public, "Would this be understandable by the man on the Clapham omnibus" is the test. Think of a typical software Eula.
When it was first used, Clapham was a lower middle class (in modern terms) area of London in commuting distance to the city (financial district) so mainly clerks, junior office workers etc. So the test was actually pretty specific to somebody who could read and write but wasn't a professional lawyer/accountant/stockbroker. Since then London property prices mean that the man on the Clapham omnibus now almost certainly IS a professional lawyer/accountant/stockbroker.