The Liouville Theorem is in fact quite easy to understand ... though a formal proof might be more complicated.
Anyway. Physicists like to look at problems in
phase space, that is, a coordinate system of spatial and momentum coordinates. In that space an object therefore usually has 6 coordinates  3 for space (x1,x2,x3) and 3 for momentum (p1,p2,p3). These values determine the future of the object  using the
Hamilton formalism one can calculate its
trajectory in phase space.
Now suppose you have a number of points in phase space like this:
p /\ .
 ....
 ......
 .....
 ..

>
x
If now one of the inner points' trajectories were to touch one of the outer points', from then on it would
evolve right along with that other point.
That means it can not cross! And in fact, two trajectories cannot even touch because there are always other trajectories in between (the coordinates are
real numbers) which would have to be crossed first!
Therefore an inner point will always remain an inner point ... the group of points may change its shape, but the area (or volume in 6d) has to stay constant because there's no place to go for the inner points. A
phase space volume behaves like an
incompressible liquid and the phase space density is constant!
PS: It seems that the theorem
ariels refers to is a different one, also called
Liouville's Boundedness Theorem.