Since the OP is a mathematician, I'll write this in a style that I would think would work for a mathematician with no specific background in physics.
Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ...
This is sort of a similar outlook to the one created by Descartes a generation earlier. We imagine space and time as imbued with preferred coordinate systems, such as Cartesian coordinates for the Euclidean plane. Although there may be some freedom to choose our coordinates, e.g., we can rotate or translate the axes, they are imagined to be rigid. There can be coordinates that are not as good, because lines don't have equations of the expected form.
In coordinates $(t,x)$, Newton's laws make predictions about $d^2x/dt^2$. Therefore we can always make a change of coordinates of the form
$$ x' = x+vt+x_0$$
$$ t' = t+t_0,$$
and because the second derivative stays the same under this transformation, the laws still work. However, there are other coordinates that we're not allowed to use, e.g., if $(t,x)$ are coordinates in which Newton's laws hold, then we're not allowed to use coordinates such as
$$ x' = x^3$$
$$ t' = t.$$
These attitudes about the absolute nature of time and space became very ingrained, but general relativity pretty much demolished them. GR takes place on a manifold, and you typically can't even coordinatize an entire manifold using one set of coordinates. Therefore we end up expressing GR (in the standard formulation) entirely in terms of tensors, in such a way that any equation remains true under any smooth change of coordinates (diffeomorphism).
The attitude is that coordinates are just names for points in spacetime, and the points don't care what we name them.
Many physical theories other than GR can also be expressed in this way, so I don't really agree with Weyl'ls characterization in detail. But it is true that some physical theories resist being cast in this way. For example, in quantum mechanics time is a parameter rather than an observable, so it can't be treated on the same footing as space, and we don't have the freedom to subject time to an arbitrary smooth change of coordinates.