Inside your capsule you would feel no gravitational forces, just as the astronauts in the International Space Station feel no gravitational forces. However you can still detect the curvature. You do this by surrounding yourself with a spherical shell of test particles that are initially at rest wrt you, and you watch them to see if they move.
The test particles nearest the planet feel slightly stronger gravity than the ones farthest away from the planet and they will move on slightly different orbits. The result is that the original spherical shape will be stretched out. By observing the change in shape of the sphere you can calculate the curvature of the space you're moving through.
The observation of gravitational lensing shows that GR works, and GR is based on the idea of curvature. Whether this counts as a proof that spacetime curvature is real depends on how philosophical you want to get about whether successful mathematical models constitute reality (whatever you mean by reality). I think most of us pragmatists would say that spacetime curvature is real.