Yes, silver reflects because of the difference of its refractive index from that of air. However, silver is different from air or vacuum in that it is a lossy medium: it is conductive, so electromagnetic waves propagating in it induce currents, thereby losing energy and decaying in amplitude. This is conveniently modeled by a complex refractive index:
$$\tilde n= n-i\kappa$$
where $\kappa$ is called the extinction coefficient. It describes how "lossy" the medium is.
With refractive index defined as such, the reflectance of an interface is given by Fresnel's equations. For normal incidence on the interface of media $1$ and $2$, the reflectance is
$$R = \left|\frac{\tilde n_2 - \tilde n_1}{\tilde n_2 + \tilde n_1}\right|^2.$$
For incidence from air on silver,
$$R = \frac{\kappa^2+(1-n)^2}{\kappa^2+(1+n)^2}$$
where $n$ and $\kappa$ refer to the refractive index of silver.
Silver is highly reflective at visible wavelengths in part because of its large extinction coefficient, and in part because of its small real refractive index.
Intuitively, since silver is highly conductive, the incident electromagnetic radiation can freely move charge within it, so it induces a current at the surface, which "shields" the rest of the material and keeps the wave from penetrating deep into silver. This current also emits radiation that appears to reflect off the surface in the direction you would expect.
