I divide this answer in three section, I, II, and III, but I give them out of order: I, III, II. The content of section II precedes the content of section III logically, but you may already have the background knowledge that is discussed in section II
Section I
In order to address that question I must first get something out of the way. In terms of GR the gravitational field and spacetime are one and the same thing, as a matter of principle.
Elaborating on that 'are one and the same thing':
The spacetime of GR is a participant in the physics taking place, as expressed by John Archibald Wheeler: "Spacetime tells matter how to move; matter tells spacetime how to curve." The purpose of that phrase is to express that in terms of GR spacetime is thought of as an entity with physical properties. It's about recognizing a reciprocity: spacetime acts upon, and is being acted upon.
Before the introduction of GR the unquestioned assumption was that spacetime and gravitational field are distinct, just as spacetime and electromagnetism are distinct. In terms of GR the concepts 'spacetime' and 'gravitational field' are unified.
Now, we don't have a word for this unified concept, which is awkward. Discussions of this topic tend to switch between 'spacetime' and 'gravitational field', suggesting a distinction that isn't there.
For the remainder of this answer I will use the expression 'GR spacetime'. GR spacetime then stands for 'the gravitational field as described by the GR equations'.
To emphasize what spacetime isn't: spacetime doesn't have parts that can be tracked through time. There is no such thing as assigning a position vector to any part of GR spacetime; there is no such thing as assigning a velocity vector to any part of GR spacetime.
That is a fundamental difference with the concept of an Aether. The whole point of an Aether theory is that such an Aether can be tracked through time.
This already addresses part of your question: I surmise you are in part asking: "Hey, there's no Aether, surely that means that there isn't anything that can be dragged at all."
(Very much an aside: it is possible that some future theory, a successor to GR, will reinstate distinction between 'spacetime' and 'gravitational field', but at present GR is what we have.)
Section III
About the expression 'frame dragging'.
Instead of 'frame dragging' this phenomenon could also have been called 'frame reorientation'.
The expression "frame dragging" is awkward to the extent that in normal use the word 'dragging' comes with a notion of dragging an object or a fluid from one position to another. Obviously in terms of GR spacetime that doesn't apply. There is no such thing as assigning a position vector to any part of GR spacetime; there is no such thing as assigning a velocity vector to any part of GR spacetime. The point is, while such vectors can't be assigned, it is still possible to make specific statements about the geometry of spacetime. See for example the two answers to the stackexchange question. How much does the curvature of space change the volume of Earth by? That is an answerable question! That demonstrates it is possible to make very specific statements.
So my recommendation would be to translate the expression 'frame dragging' to 'frame reorientation'. The word 'dragging' comes with bagage you don't need. The thing that is indispensible is that you do think of GR spacetime as having physical properties: in the vicinity of a spinning celestial body the spacetime has the property (due to the vicinity of a spinning celestial body) that the orientation of the spin axis of a spinning gyroscope undergoes change.
Section II
The gravity Probe B mission did find corroboration of frame dragging, but around the Earth the effect is so small that even for the most sensitive equipment it is still barely detectable.
In the following section I give background information on measuring change of orientation.
When a gyroscope is spinning its spin axis will remain in the same orientation with respect to the local inertial coordinate system. That is, a force is required to make the spin axis of a spinning gyroscope change orientation. In the absence of any such force the spin axis will remain in the same orientation.
(Keeping track of the orientation of the spin axis of a spinning gyroscope is only one way of tracking orientation. There are other ways of keeping track of orientation. It can also be accomplished with light. There is a widely used type of change-of-orientation sensor that is called fiber optic gyroscope.
The general name for a sensor array that keeps track of any form of acceleration (thus including change of orientation) is IMU, Inertial Measurement Unit. The frame dragging effect is fundamental and any IMU device is subject to it, not just gyroscopes. I use gyroscopes in this answer because a gyroscope is something we can see with our very eyes.)
According to GR, when a spinning gyroscope is in the vicinity of a spinning celestial body there will be a measurable change of the orientation of the spin axis, changing at a constant rate. That change is with respect to a more global coordinate system.
