Why do they have to be geometries of their own kind? For example, instead of talking about lines in spherical geometry, can’t we just imagine circles on a literal sphere in Euclidean space?
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If we have a coordinate system in say 2D Euclidean space, the distance ds between two points at location (x,y) and (x+dx,y+dy) is ds^2 = dx^2 + dy^2. That changes in non-Euclidean spaces. On the surface of the Earth, you have ds^2 = dlat^2 + cos(lat)^2 dlon^2, where dlat and dlon are the (small) differences in latitutde/longitude, and cos(lat) is the cosine of the latitude. There’s no way to come up with a coordinate system in 2D that converts those distances to the Euclidean ones, so just by measuring distances and lat/lon, you can tell you are on a curved surface. This extends to higher dimensions as well, we could have (but don’t) live on a universe where the 3D space looks like the surface of a 4D sphere. Very often, it’s easier to do math taking advantage of that fact! Because my spherical trig sucks, I still measure distance between points on the Earth’s surface by going back into 3D.
However, in real life, the situation rapidly becomes more complex. If you work in say cosmology, or with black holes, you can still write down the distances, but they no longer correspond neatly to a simple higher dimensional space where it’s easy to do math. Picture the surface of the Earth as it really is – with mountains, valleys etc. If you want to measure distances, you need to take that into account. You *could* do it by including an altitude map of the Earth, but very often what happens is that the math/physics tell you how to calculate the distance, so it would be an awful lot of work to go from that back to the 3D surface when you could always just work in lat/lon, where elevation just gets taken care of from the physics.
If this sounds complicated, it is! That’s why people don’t take courses in general relativity until they’re getting graduate degrees in physics.
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