During a collision, the padding on the inside of the helmet increases the collision time and therefore reduces the magnitude of the interaction force.  The hard shell distributes the interaction forces over a larger area and therefore decreases the pressure.

So how well does it work?

In this video clip on YouTube, Dr. Tim Gay demonstrates how a football helmet protects a Honeydew melon dropped by a person on a ladder.

Can we reproduce this feat?

We looked at another example of a hard shell distributing the force and reducing the pressure. We placed an egg into a bag filled with a mixture of starch and water.  This mixture forms a strange fluid, called a non-Newtonian fluid.  Rapidly applying a force to the fluid makes to the fluid behave like a solid.  But when the fluid is handled gently, it stays in its liquid state.  At the moment the bag containing the egg and the fluid hits the floor, the fluid immediately solidified and forms a thick, hard shell around the egg, distributing the force and reducing the pressure on the egg.  After the impulse, the fluid returns to its liquid state.