Slow In and Out has to do with the acceleration of the object being
animated. An object can't go from standstill to a speed without
accelerating. As a car stops it does so slowly and slowly speeds up.
Although the use of "slow" in this case may not be the
best word to use, the idea is simple, for example a bouncing ball.
When we bounce a ball there are only two times that the ball is not
moving for just a split second: at the top of the bounce and the bottom
of the bounce. As the ball approaches the top it slows down till it
stops (Slow In). Once the ball has stopped for that micro second it
is not moving. Then the ball starts to accelerate to the ground again
because of gravity, but it is done slowly (Slow Out).
As it turns out this type of motion is repeated for all moving under
acceleration objects. Sometimes the motion is so fast that it can
be ignored like a gunshot but with some things is would look unnatural
with out the slow in slow out.
Here is a good example of this process.
The ball on the left moves at a constant speed with no eases and no squash/stretch.
The ball in the center does slow in and out with squash/stretch.
The ball on the right moves at a constant speed, no eases, but with squash/stretch.
Now with the timing sped up, the sacing favouring the top even more so the bounce impact is snappier, you have this verion with slow in+out at the top pose, and slight squash+stretch on impact.