In order to understand asymptotic freedom, you need to be aware of the concept of renormalization. Since you want a qualitative description, just think of renormalization a modification of the coupling strengths and masses of particles at high energies. This is roughly like pushing a ball through the water; the harder you push, the more the water sticks around it and the harder it is to move. This can be modeled with Newton's 2nd law $F=ma$ by replacing the mass with a slightly larger mass $m+\delta m$, and this $\delta m$ depends on the velocity of the ball in the water.
(that discussion can be found in section 3.2 of Connes and Marcolli, "Noncommutative Geometry, Quantum Fields and Motives")
Once you have the concept of renormalization, asymptotic freedom is a property the strong force has as you scale the coupling constant to high energy. Rather then the coupling getting stronger, it gets weaker. This has major consequences for confinement - that is, bound quarks. At low energies, quarks in bound states are forever bound - it becomes harder and harder to pull them apart the further apart you pull them. At high enough energies (say, colliding two protons at 7 TeV like the LHC) the quark coupling gets small and quarks are essentially free and unbound. It should be easy to see how this would change the cross section.
As a sidenote, only the strong force is asymptotically free. The E/M and weak force become stronger as the energy gets higher. In addition, it is important to realize that we cannot solve problems involving the strong force at low energies (if you could, the Clay Mathematics Institute would give you $1 million!). Once they are at high energies, the strong coupling is weak so QCD acts quite a bit like QED.