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We studied the statistical properties of the HI-to-H$_2$ transition observed in absorption in the local diffuse and multiphase ISM to identify the physical processes controlling the probability of occurrence of any line of sight. The turbulent diffuse ISM is modeled using the RAMSES code, which includes detailed treatments of the magnetohydrodynamics, the thermal evolution of the gas, and the chemistry of H$_2$. The impacts of the UV radiation field, the mean density, the turbulent forcing, the integral scale, the magnetic field, and the gravity on the molecular content of the gas are explored through a parametric study covering a wide range of physical conditions. The statistics of the HI-to-H$_2$ transition are interpreted through analytical prescriptions and compared with the observations using a modified and robust version of the Kolmogorov-Smirnov test. The results of one simulation, convolved with the distribution of distances of the observational sample, are able to explain most of the statistical properties of the HI-to-H$_2$ transition observed in the local ISM. The tightest agreement is obtained for a neutral diffuse gas modeled over ~200 pc, with a mean density of $1-2$ cm$^{-3}$, illuminated by the standard interstellar UV radiation field, and stirred up by a large-scale compressive turbulent forcing. Within this configuration, the 2D probability histogram of the column densities of H and H$_2$ is remarkably stable and is almost unaltered by gravity, the strength of the turbulent forcing, the resolution of the simulation, or the strength of the magnetic field $B_x$. The weak effect of the resolution and our analytical prescription suggest that the column densities of HI are likely built up in large-scale WNM and CNM structures correlated in density over ~20 pc and ~10 pc, respectively, while those of H$_2$ are built up in CNM structures between ~3 pc and ~10 pc.