The core is simply the mid section of your body or the muscles that comprise the torso and their corresponding ligaments and tendons. It is in this section that all sport and functional movement originates. It is how you have control over your center of gravity which has a direct impact on balance. It is because of this responsibility that core exercises are critical for a healthy body. Do not misunderstand mean, core exercises are like vegetables. They will have a significantly positive impact on your overall health, but many people somehow live a long healthy life without them.
For the most part, exercises that require the core section to be incorporated (i.e. a plank hold) will build the strength of the back and pelvis along with tightening and firming the mid section. This is different than losing belly fat. Core exercises also build the stabilization necessary to function more comfortably. It is better to be able to sneeze or pickup your child without discomfort, than always feeling like your back is going to 'go out.'
So what should you do beyond basic crunches and leg lifts? The easiest way to figure out the anatomy of your core is to begin by coughing. A deep cough will allow you to feel the transverse abdominis (the most center section of the abs). In order to activate the muscles comprehensively throughout your core (meaning from the inside wall to the outside wall of muscular layers) you want to do exercises that require other sections of your body to be involved. It is a misconception that the center section of abs (rectus abdominus) be the primary muscle working. It is better to incorporate exercises that engage all the other areas of the torso and abdominal family. Although a basic crunch is valuable it is not going to recruit many other areas of the body other than the center section of abs. There is nothing wrong with doing variations of the crunch. But anytime you are causing other muscles to work (meaning: muscles to fire a signal and contract), you are improving how your muscles work together during movement. This will generally help prevent injuries and improve your overall balance and strength.
Let's take a quick look at the core section (see featured pic). The main rectus abdominus bends or flexes the spine or torso, and the obliques, which are located along the side rotate the torso.
The exercises in the video provide a basic crunch with some resistance elements. It is the addition of the xering resistance tubing that builds additional strength and provides a better core workout.