What software is used for aerodynamic wing shape optimization?
If you follow the AIAA publication number (AIAA-2014-0567) given in the first frame of the linked video, you will find this article which explains in detail that the optimisation shown in the video was done on MACH:


  These tools are components of the framework for multidisciplinary
  design optimization (MDO) of Aircraft Configurations with High
  fidelity (MACH). MACH can perform the simultaneous
  optimization of aerodynamic shape and structural sizing variables
  considering aeroelastic deflections.


If you now google for MACH, you will find this article by Gaetan Kenway which explains in detail how the software works. But this is not what you want, my impression is you need a simple to use software which can be downloaded and run within minutes.

My recommendation would be XFLR5 which incorporates the XFOIL airfoil solver and adds modules for designing and calculating the whole aircraft. YouTube has a variety of tutorials available. Is that what you want?

However, a basic understanding of aerodynamics and the possibilities and limitations of the algorithms used will be very helpful - a good engineer should be able to think both like a scientist and a programmer.