For instance, imagine you animated an elephant dropping from a building. As the elephant fell, you would stretch and thin the elephant's shape to convey that the momentum of falling is pulling on the elephant's mass and dragging it out into an elongated shape. This helps to convey momentum, and also the force exerted by the air against the elephant as it falls. Without stretching its shape, the fall seems almost leisurely, at a steady speed. Increasing the stretch as time passes can convey momentum gained over the fall.
When the elephant lands, though, you would squash its shape - and not just back into its normal proportions. The impact against the ground would push hard against the elephant's mass, and as the stretched shape strikes the hard surface, gravity and momentum would push down on the elephant to flatten it out of its usual shape. Without squashing the shape, the elephant would appear not to have any mass, and the animation would seem unrealistic. The amount of squash can also help convey the force of impact.
The amount of squash and stretch can also be used as a subtle way to convey differences in mass, texture, and substance. A ball and an elephant would hit the ground differently; the ball wouldn't squash as much as the elephant because it's not as heavy. A feather or a piece of paper wouldn't squash much all, because they're so light; any squash would be very gentle - but they might be stretched as they're pushed along by wind or another force. When animating ethereal objects, you can often show their ephemeral nature by having them skim along surfaces without distorting.
One reason squash and stretch are so important is because they occur in real life. Our muscles squash and stretch every time we use them. Objects distort when force is exerted on them. For example, watch an episode of Mythbusters; especially the slow-motion sequences of something striking ballistics gel. Or, for more fun, of Jamie slapping Adam. When something strikes the ballistics gel, it's the gel that squashes and stretches as it absorbs the impact; the force is transferred and distorts it out of shape. When Jamie slaps Adam, his cheek first squashes as it's struck, then stretches out with momentum as his head whips to the side. One key point of realistically animating different things is studying them to see the differences in their squash and stretch in different situations.
