What this means is that instead of having to keep track of all the data associated with each object globally and differentiate between them, you can let each object look after itself.
For example, imagine that you've got a bunch of sprites moving around onscreen, and you want to move them all at different speeds. Obviously you can't have a single global value for speed, because that would be the same for all sprites. It would be possible to keep a list of all the different speeds, but then you'd have to search through it and match the right speed with the right sprite.
By using child objects to control the sprites, and declaring a property variable for speed in the parent script, each object can maintain its own value for speed, and the handler which moves them about can treat speed as a single set value (assuming that handler is also in the parent script, which is, after all, the sensible place to put it).
Property variables are very like sprite or cast properties in this respect: each sprite has its own rect, ink, puppet etc, but they're all referred to by the same name. Just like other properties, a child object's property variables can be determined by handlers outside the object by using the keyword "the" :
put the speed of someObject