What you have to keep in mind is that General Relativity is built upon the Equivalence Principle. The strong version of this principle is that locally, whichever spacetime you're working with will be <PERSON> spacetime, i.e. it will be locally flat. That's the key aspect of the (strong) Equivalence Principle, it is local. Therefore a local observer will not be able to tell if they're present in a curved spacetime, this observer will have to make non-local measurements in order to tell if their spacetime is curved.
Take the gravitational field caused by the Earth. Locally, a free falling observer A will be in inertial motion (considering that their mass is small compared to the Earth's mass) since they're following a geodesic. However, consider another free falling observer B, and put A and B relatively near each other but far from the surface of the Earth (for the sake of simplicity, let's suppose they're equally far away from the Earth's surface). You'll notice that they will approach each other as they're falling towards the Earth, almost as if there's a "force" that's pulling them together. This is a tidal force that is present due to the curvature the Earth induces on spacetime. 562
In contrast, if A and B are falling near the Earth's surface, they will not seem to approach each other because, locally, there is no apparent curvature and spacetime is flat.
As an extreme example, consider you're falling towards a sufficiently small black hole, so that your soles are facing towards the event horizon. Since the curvature of this black hole is so extreme, as you're free falling towards it, the nearer you are to it the stronger the pull is on your feet than it is on your head, i.e. there will be a tidal force that will eventually stretch you violently and you'll be spaghettified before you reach the horizon. In this example, you're not a local observer since the curvature is so extreme, i.e. locally you're not in <PERSON> spacetime.
Edit: I'd like to add to your last point that matter is not external to spacetime, it is embedded in it. The reason why this curvature seems to leave this matter unchanged, is because gravity is very weak compared to the other three forces. You would need to have an extreme gravitational field in order for matter to be affected by it. That's why spaghettification happens. The electromagnetic forces that hold you together aren't strong enough to hold against the tidal forces caused by the black hole's curvature. 43