I think he means it in the sense that if you take relativistic equations, and substitute infinity in for c (or, more rigorously, take the limit as c goes to infinity), you will get Newtonian equations. Thus the behavior of objects at small speeds is roughly Newtonian, because c is already well on its way to infinity compared to those speeds; conversely, when an object is traveling at a rate of 0.1c, it matters greatly that c is finite.