How does centrifugal force work? I know what centrifugal force is, but how does it work? Why are things forced to the outside?
 A: Because things want to go in a straight line.
Imagine swinging an object around your head on a string. At any moment the object wants to go straight ahead (ie on a tangent) but it can't because of the string. 
It's as if the string was pulling it back in toward you - that's balanced by a force pushing it outward = centrifugal force.
For a more technical explanation try physics.stackexchange.com, but expect an arguement about whether centrifugal force exists!

A: Centrifugal force is not a real force in Newtonian mechanics. Whenever a frame of reference is accelerated (except the cases where the acceleration is due to gravity) w.r.t. an inertial frame of reference, an observer situating in that reference frame experiences a "force" in the opposite direction of the acceleration. 
For an outside inertial observer this is nothing but the inertia of the objects situated in the accelerated frame. Since in Newtonian mechanics strictly the descriptions of only the inertial observers are legal hence the "force" experience by the objects in the accelerated frames is not a real force. It is called a pseudo force.
A rotating framed of reference is also an accelerated frame where the acceleration is directed towards the center. It is called the centripetal force. As always, as described above, a "force" will be experienced by the objects situated in that rotating frame opposite to the centripetal acceleration i.e. away from the center. This is called the centrifugal "force".
Obviously it is not a real force from the point of view of an outside inertial observer, who is the legal observer in the Newtonian mechanics. For this observer it is simply the inertia of the objects.
Therefore we call this centrifugal force as a pseudo force and not a real force.
From General Relativistic considerations however all observers are equivalent no matter how they are moving and all accelerations are equivalent to gravitational field in a short enough scaled. Therefore for an infinitesimally small rotating frame the centrifugal "force" can be thought of as a gravitational force! This directly follows from the equivalence principle.
A: The real force at work is centripetal force, or a force pushing inwards.
Imagine you have a bucket on a string, and you swing that around in a circle:

As you swing the bucket, it travels in a circle. The red line shows the path the bucket takes. In order to make it swing like this, you have to apply a constant force on the rope -- this is the green arrow in the image. At any given moment in time, the bucket wants to travel straight -- the blue line in the diagram. By applying the centripetal force, the inward force, you change the motion from straight to the circular motion (the red line).
Because the contents of the bucket always want to go straight, and the force you apply always make them change direction, there seems to be an "outward" or "centrifugal" force "pushing" the contents against the side of the bucket. But it's an an illusion -- it's really just the momentum of the bucket and it's contents.
A: Centrifugal force is a particular example of a fictitious force.  It is introduced so that Newton's second law holds in a rotating reference frame.
Newton's second law says
$$F = ma$$
This means that whenever we find an object accelerating (speeding up, slowing down, turning, or some combination), we can look around and find a physical reason why this happens.  For example, a dropped stone accelerates towards the Earth, and this is due to Earth's gravity; if we drop the stone far from Earth, it won't fall.  Your car turns a corner.  This happens due to friction with the road.  If the road were perfectly slick, the car would simply slide.
Newton's second law holds in an inertial reference frame.  It is simply a fact that such reference frames exist, and that they are all related to each other by moving past each other at constant velocities.  (This becomes more complicated in general relativity, but that is not a major concern in everyday situations.)
However, suppose you are in a train that begins accelerating forward (from the stationary track's point of view), and you are looking out the window at a ball sitting on the sidewalk nearby.  From your reference frame in the  train, the ball is accelerating backwards.  However, there is no obvious source of a force on the ball that would make it accelerate backwards.  This means that in an accelerating frame, Newton's second law doesn't work.
Sometimes we would still like to do physics in such an accelerating frame, so we simply invent a new force, called a fictitious force, and say that the ball has a fictitious force of just the right amount needed to give it the acceleration we observe.  Since the ball's acceleration is $a_b = -a_t$  with $a_t$ the acceleration of the train in an inertial frame, we need to introduce a fictitious force
$$F_{fict} = -m a_t = m a_b$$
That way, Newton's laws still work and we can do physics as normal as long as the train's acceleration stays the same.  We could, for example, play billiards in the accelerating train, noticing the the balls have curved trajectories across the table, and these curved trajectories would be perfectly explained by a fictitious force $-ma_t$ acting on each ball (with $m$ changing for balls with different masses).  Keep in mind that the fictitious force points in the opposite direction of the train's acceleration.  If the train accelerates forward, the ball appears to accelerate backwards, so the fictitious force must point backwards.
Another type of accelerating frame is a rotating reference frame, for example a carousel.  On the carousel, every part is accelerating towards the center (see Josh's answer).  Therefore, to do physics in this frame, we must introduce a fictitious force
$$F = -m a_c$$
as before.  $a_c$ is the acceleration of the carousel at any point.  Because this acceleration points in towards the center of the carousel, the fictitious force points the opposite direction - out towards the edge.  This fictitious force is called the centrifugal force.
Introducing the centrifugal force lets us do physics from the point of view of the rotating carousel, with the caveat that we can only handle statics this way.  If things are actually moving on the carousel, we need to include the Coriolis force, which pushes things sideways.  (See derivation  here  or some discussion of it in my answers here or here.)
As for whether the centrifugal force is "real", it depends on what that means.  In an inertial frame, each force can be traced back to some physical interaction like the exchange of a photon.  That's not true for the centrifugal force.  This is the essential difference people are referencing when they say the centrifugal force and other fictitious forces "are not real".
