A spline is a curve constructed from several joint polynomial equations. The higher order splines are splines made up of higher order polynomials. As a general rule, the higher the order, the more fluid the curve (in a way), the linear splines being a series of straight lines.
Here are some examples or different orders (from the Wikipedia page on Bézier curves, a type of spline).
These GIFs usually show how these curves are constructed and
higher order = more control points. In the example of Amit, these checkpoints are consecutive points along the way. A higher order spline therefore means that there is a softer interpolation along a larger set of these points at a time.
In other words, instead of only looking at the next point (linear), the next two points (quadratic) or the next three points (cubic) could be considered. This allows a more natural movement of the entities in your game along the path that has been built – the underlying logical path does not change, just the actual motion on the screen.
There is more information on the Wikipedia page on how to build these curves.