I think this is the main difference between mathematicians, who are very much emphasizing the geometric aspects of spacetime. This has some justification since modern physics hinges on geometric concepts in a very general sense, discovered by Riemann and Klein in the 19th century, namely the importance of symmetries (of spacetime and even abstract "flavor spaces" in QFT). This point of view in physics was one of the most important and usually overlooked breakthroughs in Einstein's paper on SR of 1905, where the first sentence can be read as a research program going on until today, namely to figure out the basic symmetries of the physical laws.
However, one can also overemphasize the geometrical aspects and forget that besides the elegant formulations in terms of geometric objects (differentiable manifolds, affine Euclidean and pseudo-Euclidean, Riemann- and pseudo-Riemann, varous fiber bundles,...) we still do physics, and physics is about what you can really observe (in a very broad sense, from naive looking at things with our senses to high-precision quantitative measurements with very tricky technology). Usually we don't realize it anymore, but any measurement (more or less tacitly) uses and introduces a reference frame. This can be simply the edges of our laboratory or the geometric setup of a particle detector or some fancy optical device (like a Michelson-Morley interferometer) etc. etc. A reference frame is somehow defined by real physical objects, be it a human being with his senses looking at a phenomenon or any fancy measurement device invented to discover accurate quantitative facts about nature that are not directly "detectable" by our senses.Now, in a gedanken experiment a Rindler observer can be seen as sitting in a rocket that is accelerated with constant proper acceleration. Now to cover some finite part of Minkowski space you need a whole family of such (pointlike) observers. This family trajectories is defined by a real spatial parameter ##\xi## and temporal parameter ##\tau## according to
$$t=\xi \sinh \tau, \quad x=\xi \cosh \tau.$$
The four-velocity of each observer (labeled by the parameter ##\xi##, with ##\tau## the parameter of the trajectory) is given by
$$(u^0,u^1)=\frac{1}{\sqrt{\partial_{\tau} t^2-\partial_{\tau} x^2}}(\partial_{\tau} t,\partial_{\tau} x)=(\cosh \tau,\sinh \tau).$$
As you see, each observer in the family is clearly accelerated, because ##u^1## is not constant. The proper acceleration for each observer is
$$\alpha(\xi)=\frac{\mathrm{d} u^1}{\mathrm{d} t}=\frac{\mathrm{d} u^1}{\mathrm{d} \tau} \left (\frac{\mathrm{d} t}{\mathrm{d} \tau} \right)^{-1}=\frac{1}{\xi}.$$
The Minkowski pseudometric reads in the new local coordinates ##(\tau,\xi)##
$$\mathrm{d} s^2=\mathrm{d}t^2-\mathrm{d} x^2=(\mathrm{d} \tau \partial_{\tau} t+\mathrm{d}\xi \partial_{\xi} t)^2-(\mathrm{d} \tau \partial_{\tau} x+\mathrm{d} \xi \partial_{\xi} x)^2=\xi^2 \mathrm{d} \tau^2-\mathrm{d} \xi^2.$$
Obviously at ##\xi=0## we have a coordinate singularity. Since for any ##\xi>0## we have
$$\frac{t}{x}=\tanh \tau$$
for ##\xi \rightarrow 0## we have ##\tau \rightarrow \pm \infty## and thus the limit ##\xi \rightarrow 0^+## defines the Rindler wedge ##x=\pm t>0##, which is an event horizon for the Rindler map of the so defined part of the Minkowski space. For a picture, see the Wikipedia article
https://en.wikipedia.org/wiki/Rindler_coordinates#Relation_to_Cartesian_chart
There you clearly see, that each Rindler observer is in hyperbolic motion with his proper acceleration ##1/\xi## and thus accelerated.Note that in the Wikipedia article they have a (to may taste a bit confusing) convention our ##(\tau,\xi)## are their ##(t,x)## and our ##(t,x)## are their ##(T,X)##. We have also set conveniently their ##g=1## and used the west-coast instead of the east-coast convention.
The Christoffel symbols are not vanishing, which is another hint that the Rindler coordinates are not locally inertial:
$$\Gamma^0_{01}=\Gamma^{0}_{10}=1/\xi, \quad \Gamma^{1}_{00}=\xi,$$
and all other Christoffel symbols are vanishing.
The geodesics of the spacetime are no straight lines wrt. the Rindler frame, and thus again we see that the Rindler frame is not inertial. Of course, the geodesics are straight lines wrt. any inertial frame.
