Why has the speed of light in a gravitational field always the same value $c$ when measured locally?
More precise: When you are far away from any masses, that means when the spacetime is flat, and you measure the speed of a light ray that passes you, you will always get the same value $c$. Now suppose you are in a static gravitational field at a fixed position, and do the same. Why do you also measure the same value $c$? (Please note: You are not in a free falling system where gravity vanishes)
As far as I understand your question, i think it's simply because gravity can only bend the path of a light ray, so if you measure speed of light $c$ in a flat space and time or a curved one (by some mass and of course gravity), there'll be no change in speed of light and more precisely as per Einstein's two main principles in relativity, speed of light always remains constant.
When you are far away from any masses, that means when the spacetime is flat, and you measure the speed of a light ray that passes you, you will always get the same value c.
Yes.
Now suppose you are in a static gravitational field at a fixed position, and do the same. Why do you also measure the same value c?
The correct question is "why not"?
We're all sitting on a geodesic, and that's the same for everyone around. Everything that you might measure, and anything you might measure with, all share the same physics.
So when you're on the flatter spacetime then you're using a flat-ruler and flat-watch to measure the flat-light. And when you're on a curved spacetime then you're using a curve-ruler and curve-watch to measure curve-light. So, one would normally imagine that you'd get the same results.
This is the whole idea of relativity; your local conditions define physics and since measurements are physical, they're defined by your local conditions too. Everyone is subject to those conditions, so it all ends up looking the same in the end.
