I can't run with the big dogs of statistics who have answered before me, and perhaps my thinking is naive, but I look at it this way...
Imagine you're in a car and you're going down the road and turning the wheel left and right and pressing the gas pedal and the brakes frantically. Yet the car is moving along smoothly, unaffected by your actions. You'd immediately suspect that you weren't in a real car, and perhaps if we looked closely we'd determine that you're on a ride in Disney World. (If you were in a real car, you would be in mortal danger, but let's not go there.)
On the other hand, if you were driving down the road in a car and turning the wheel just slightly left or right immediately resulted in the car moving, taping the brakes resulted in a strong deceleration, while pressing the gas pedal threw you back into the seat. You might suspect that you were in a high-performance sports car.
In general, you probably experience something between those two extremes. The degree to which your inputs (steering, brakes, gas) directly affect the car's motion gives you a clue as to the quality of the car. That is, the more of your car's variance in motion that is related to your actions the better the car, and the more that the car moves independently of your control the worse the car is.
In a similar manner, you're talking about creating a model for some data (let's call this data $y$), based on some other sets of data (let's call them $x_1, x_2, ..., x_i$). If $y$ doesn't vary, it's like a car that's not moving and there's really no point in discussing if the car (model) works well or not, so we'll assume $y$ does vary.
Just like the car, a good-quality model will have a good relationship between the results $y$ varying and the inputs $x_i$ varying. Unlike a car, the $x_i$ do not necessarily cause $y$ to change, but if the model is going to be useful the $x_i$ need to change in a close relationship to $y$. In other words, the $x_i$ explain much of the variance in $y$.
P.S. I wasn't able to come up with a Winnie The Pooh analogy, but I tried.
P.P.S. [EDIT:] Note that I'm addressing this particular question. Don't be confused into thinking that if you account for 100% of the variance your model will perform wonderfully. You also need to think about over-fitting, where your model is so flexible that it fits the training data very closely -- including its random quirks and oddities. To use the analogy, you want a car that has good steering and brakes, but you want it to work well out on the road, not just in the test track you're using.