Meaning of Einstein's condition on the metric tensor Quote from Nothing but coincidences: the point-coincidence and Einstein’s struggle with the meaning of coordinates in physics by Marco Giovanelli:

On November 11, 1915, Einstein returned to a set of generally covariant vacuum field equations that he had introduced in 1913:$$R_{\mu\nu} = \kappa T_{\mu\nu},$$where $R_{\mu\nu}$ is what we now call the Ricci tensor. He replaced the restriction imposed by the conservation laws with the requirement that the determinant of the metric satisfied the condition $\sqrt{-g}=1$.

What is the physical meaning of this condition that Einstein imposed on the metric tensor?
 A: I never really understood the line of thought behind Einstein's struggles with Riemannian geometry and general covariance, so this answer is given without the underlying context of the question.
Let $M$ be an $m$ dimensional manifold with local coordinates $x^\mu$ and line element $\mathrm ds^2=g_{\mu\nu}\mathrm dx^\mu\mathrm dx^\nu$. Let $$ \rho=\sqrt{\left|\det (g_{\mu\nu})\right|}. $$
It is known from Riemannian geometry that if an infinitesimal parallepiped with vertices $(x^1,\dots,x^m)$, $(x^1+\mathrm dx^1,\dots,x^m)$, ..., $(x^1,\dots,x^m+\mathrm dx^m)$ is given, then the volume of this parallelepiped is $$ \mathrm dV(x)=\rho(x)\mathrm dx^1\dots\mathrm dx^m. $$
The coordinate condition $\rho=1$ then means that $$ \mathrm dV(x)=\mathrm dx^1\dots\mathrm dx^m, $$i.e. the coordinate volume coincides with the invariant/geometric volume.
This simplifies some formulae, for example if $f=f(x)$ is a function, then its integral over a coordinate region $D$ is calculated as $$ I=\int_Df(x)\mathrm dx^1\dots\mathrm dx^m $$with no additional factor here, and if $X^\mu$ is a vector field, its divergence is calculated as $$\mathrm{div}(X)=\frac{1}{\rho}\partial_\mu(\rho X^\mu)=\partial_\mu X^\mu, $$i.e. the divergence reduces to the Euclidean formula.
Geometrically, the condition $\rho=1$ means that the grid of coordinate lines is such that the parallelepipeds formed by the intersection of the coordinate lines have unit volume.

As $\rho=1$ is a coordinate condition, physical meaning (when $M$ is the four dimensional Lorentzian spacetime) can be ascribed to it only if it represents a nontrivial restriction on the nature of spacetime.
Consider for example another coordinate condition $$ g_{\mu\nu}=c_{\mu\nu}, $$where the $c_{\mu\nu}$ are constants. If this is the case, then the metric can be transformed into pseudo-Euclidean form by a linear transformation of the coordinates. Indeed, a general pseudo-Riemannian space does not admit such coordinates, and the integrability conditions for the existence of such coordinate systems is that the Riemann curvature tensor should vanish.
Hence, the physical meaning of $g_{\mu\nu}=c_{\mu\nu}$ is that the spacetime is flat.
However the condition $\rho=1$ is devoid of such a meaning, since every pseudo-Riemannian space admits an atlas of coordinate charts which satisfy this condition.
This can be seen by noting that if the function $$\phi(x)=\phi(x^1,\dots,x^m)$$ is given in a neighborhood of $x_0$ and does not vanish anywhere, then it is possible to make a coordinate transformation $$ \bar x^\mu=\bar x^\mu(x^1,\dots,x^m) $$ in a neighborhood of $x_0$ such that the Jacobian of the transformation is $\phi$. For example by taking $$ \psi(x^1,\dots,x^m)=\int_{x_0^1}^{x^1}\phi(s,x^2,\dots,x^m)\mathrm ds, $$ then $$ \phi=\partial_1\psi, $$ and if we define $$ \bar x^1=\psi(x^1,\dots,x^m),\quad \bar x^2=x^2,\dots,\bar x^m=x^m, $$ we have $$ \left(\frac{\partial\bar x^\mu}{\partial x^\nu}\right)=\left(\begin{matrix}\phi & \partial_2 \psi & \cdots & \partial_m\psi \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1\end{matrix}\right), $$ and the determinant of this matrix is $$\det\left(\frac{\partial\bar x^\mu}{\partial x^\nu}\right)=\phi. $$
Hence  if we take $\phi:=\rho$, we get $$ \bar\rho=\det\frac{\partial x^\mu}{\partial \bar x^\nu}\rho =\frac{\rho}{\phi}=1. $$
Since every pseudo-Riemann manifold can satisfy the coordinate condition $\rho=1$, it does not restrict the spacetime manifold in any way and therefore has no intrinsic physical meaning.
