(I posted a similar question on math.se.)
In information geometry, the determinant of the Fisher information matrix is a natural volume form on a statistical manifold, so it has a nice geometrical interpretation. The fact that it appears in the definition of a Jeffreys prior, for example, is linked to its invariance under reparametrizations, which is (imho) a geometrical property.
But what is that determinant in statistics? Does it measure anything meaningful? (For example, I would say that if it is zero, then the parameters are not independent. Does this go any further?)
Also, is there any closed form to compute it, at least in some "easy" cases?