What's the physical content in the invariance of spacetime interval in GR? Spacetime interval in one co-ordinate system is given by : $$g_{\mu \nu} dx^{\mu} dx^{\nu} \tag{1}$$
$dx$ is some infinitesimal displacement vector between two events.
Spacetime interval after a change of co-ordinate system is give by the algorithm : Change the basis of the matrix $g_{\mu \nu}$, Change the basis of the vector $dx^{\mu}$, and then calculate the same quantity as in $(1)$.
So of course a change of basis leaves the spacetime interval invariant. This is a purely mathematical fact.
What's the physical content here? I understand the physical content of spacetime interval in $SR$, because there the four components of $dx^{\mu}$ refer to actual space and time measurements using clocks and sticks.
In GR however, $dx^{u}$ is more abstract as the four indices don't refer to space and time measurements but to generalised co-ordinates....
In fact, even if we assume that the four indices of $dx^{\mu}$ in GR refer to actual spacetime-measurements by an observer, then a "change to another generalised co-ordinate system" need not mean a "change to another physical situation". Let me explain.
Suppose a GR observer measures $dx^{\mu}={dt, dx, dy, dz}$ using sticks and clocks, and calculates $(1)$. Then we switch to another co-ordinate system : $(dt, r, \theta, \phi)$, and then we calculate $(1)$ again and find it to be invariant. But this is no surprise as the change of co-ordinates was purely mathematical. The "new co-ordinates" refer to the same observer using different variables to parametrise spacetime.
In SR, the invariance of $(1)$ relates spacetime measurements made by observers in two different physical situations. In GR, this isn't the case either.
 A: At any spacetime point $x$, you can always go to locally inertial coordinates where $g_{\mu\nu}$ at $x$ is simply the Minkowski metric, $\eta_{\mu\nu}$, and the Christoffel symbols vanish. Hopefully it is clear that in these coordinates, in a small neighborhood around $x$, the interpretation of the invariant spacetime interval in a neighborhood of $x$ is the same as the interpretation of the invariant spacetime interval in special relativity. The fact that the interval is invariant under general choices of coordinate means that we can calculate the interval in any coordinate system, without having to explicitly go to locally inertial coordinates.
In particular, whether the separation between two nearby points is spacelike, timelike, or null can be calculated in any coordinates. Since the value of the interval is invariant, we know without having to do any calculations that we always could go to a locally inertial frame where we can use our intuition about spacelike, timelike, and null separations from special relativity.
A: 
What's the physical content here?

The physical content is that measurements of distance and time do not depend on the choice of coordinates.

even if we assume that the four indices of  in GR refer to actual spacetime-measurements by an observer,

No, that is not a valid assumption. The coordinates are not measurements, they are just labels. The actual spacetime-measurements are invariants like $g_{\mu\nu}dx^\mu dx^\nu$
