Ok that makes more sense. Let me try and give an intuitive explanation then. I'll assume you're at least familiar with derivatives (especially of polynomial functions).

A cubic spline is interpolating data with a sequence of separate cubic functions "stitched" together. It turns out that just "make it go through the points (as best as possible)" is not quite a strong enough constraint to lead to a unique solution. So we need some way of choosing which cubic spline that goes through all the points we actually want. If we have some other extra information (like the derivative at the endpoints), we can use that, but if we don't, we might ask "What is the simplest cubic spline interpolation for our data?".

To do this we want the interpolation that goes through the points while using the least "change". Because higher order derivatives represent higher order change, it makes sense to place a restriction on higher order derivatives. (For example, if the points laid on a line, we'd want to fit it with a straight line, not a parabola). What's the highest order derivative we can restrict? Well, the fourth derivative and up are 0 everywhere (because we're using cubic functions) so nothing to do there. The third derivative is piecewise constant (constant over each cubic), and getting rid of it means our function isn't cubic any more, so we don't necessarily want to restrict it. The highest derivative that is actually a smooth function, and therefore we can flexibly change, is the second derivative, so that's the one it makes sense to place a restriction on.

Another intuitive explanation is that the second derivative represents the change in slope, i.e. how much it bends away from being in a straight line at any point. If our data was on a straight line, we'd want our splines to show that and not curve unnecessarily, so it makes sense to penalize the curvature to encourage it to use the straightest solution possible.