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Statistics`Library` does not correspond directly to a package path, so it cannot be loaded with Get or Needs. Rather, this contents of this context come from

FileNameJoin[{
 $InstallationDirectory, "SystemFiles", "Kernel", "SystemResources", $OperatingSystem,
 "Statistics", "Library.mx"
}]

which is loaded automatically at startup. Loading this file populates the context, but does not add anything to the context path.

It is straightforward to change the context path, however:

AppendTo[$ContextPath, "Statistics`Library`"]

Now:

DistributionDimensionality@NormalDistribution[] (* -> 1 *)