A spline is a curve built from a number of conjoined polynomial equations. Higher order splines are splines built of higher order polynomials. Generally speaking the higher the order, the more fluid the curve (sort of), with 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 show generally how these curves are constructed, and
higher order = more control points. In Amit's example, those control points are consecutive points along the path, so a higher order spline means that there is a smoother interpolation along a larger set of those points at a time.
Put another way, instead of looking at just the next point to move toward (linear), it might look at the next two points (quadratic) or next three points (cubic). This allows for more natural movement of your game entities along the path that has been constructed - the underlying logical path doesn't change, just the actual movement on screen.
There is more information on the Wikipedia page about how to construct these curves.