They mean the act of putting your weight on your equipment. If you're climbing up by pulling on the rock, and resting on ledges, that's fine, but taking a rest by sitting back in your harness, or pulling on the rope to gain height is not.
Climbing is an one of the more logical sports, but like all sports, arbitrary rules have to be set if you want to accurately compare achievements between two people. It's just a way to make sure everyone knows what you mean when you say you've 'done' a route.
That said, I recommend ignoring this aspect of climbing -- fulfilling personal goals is a lot more rewarding for everyone than trying to compare yourself to others.