Physically, the maximum speed (velocity is a vector! Speed is its magnitude, which is what you're talking about) happens when the object that's moving passes by the origin and that is the point of zero displacement. It's also the point of zero acceleration because the restoring force behind the motion is zero at that point (and acceleration is force divided by mass).
Incidentally, the minimum speed happens at the opposite situation, namely, at the extreme ends of the motion. Since those are the extreme ends, the particle can't move beyond those and has to momentarily stop there (hence, zero speed). Those points are also where the acceleration is largest in magnitude, pulling the particle back towards the origin.