Graphically we can illustrate this concept very easily as follows:
Related to this are the observations that financial time series have a multi-fractal nature, obey power law distributions (pdf), struggle with volatility clustering and all sorts of other nasty (from a modelling point of view, anyway) features.
The question is why?
There is a recent paper out that seems to explain this in a nice intuitive manner. Leverage + some simple assumptions naturally leads to fat tails... essentially borrowing money to invest causes investor's behaviour to change as they have to repay the debts and interest on money borrowed to invest with. This means that investors need to sell into falling markets, thereby accelerating the down moves, rather than being able to take contrarian views which would serve to counter the down moves and make the distributions more "Normal".
Remove the leveraged buying and the fat tails disappear.
This is one of the best explanations of "why" that I have seen.