The idea of relativity is that the laws of nature are independent of the reference frame.
And the basic for gravity in GR is the equivalent principle. That means: you can think as the surface of the planet was accelerating (upwards) all the time. That is (almost) all. GR doesn't try to explain why that happens.
But there is a little more than that principle. Planets are spherical, and gravity is stronger close the surface than at a mountain. The direction of that acceleration also changes for different points of the sphere.
So, the vector field of accelerations is not constant. That is the meaning of the space-time curvature.
If a spaceship is drifting in the space, and see a planet orbiting it, the crew (even while floating around the ship) knows that they are in a gravitational field, because otherwise all external objects should follow a straight line. If it is not the case there is space-time curvature.
So, space-time curvature is not a cause for gravity, but a way that floating crew in the orbiting space station agree with fixed crew in the base station that both are under Earth gravitational field.