Static spacetime A static spacetime is a spacetime that has hypersurface orthogonal timelike Killing vector field.
I know that it guarantees that the metric is invariant under time reversal and that the metric is time-independent. 
My question is that, why it is "static"? Is there an explanation like stationary metric that a stationary metric means the metric doesn't change with time?
 A: This will be a very handwaving explanation since the questions seem to look for intuition. I will not attempt at defining every word precisely, since the terms have already been well-defined by you.
Stationary spacetimes are stationary in the sense that they look the same at every instant of time, but this doesn't mean that nothing is going on. For example, the Kerr metric is stationary despite describing a rotating black hole.
Static spacetimes are static in the sense that this does not happen. The Kerr metric is not static. You not only need things to stay looking the same, but you also go a step further and demand that nothing is going on. Things not only look the same but are the same. You don't have rotating stuff.
It is worth mentioning that, due to Frobenius theorem, hypersurface orthogonality of a congruence of geodesics is equivalent to the family of geodesics not twisting (check e.g. Wald's book Chap. 9 or Wikipedia for the notion of twist/vorticity of a congruence of geodesics). Hence, asking for hypersurface orthogonality is pretty much equivalent to requiring that the stationary observers are not rotating.
Remark: as I said, this is a very handwaving explanation. Do not attempt to take the terms I used in a very serious way. I did not attempt at making this explanation completely unambiguous. The rigorous definitions are unambiguous, but I tried only to provide some intuition behind the terminology.
