We know there is no aether, so what is being dragged in frame dragging?

In frame dragging, what is exactly being dragged, the gravitational field or the fabric of spacetime itself?

Suppose that we have a set of three orthogonal gyroscopes and three orthogonal linear accelerometers, also called an inertial measuring unit (IMU), and that they are all exquisitely precise. Suppose further that we mount a telescope to this IMU and ensure that the telescope neither rotates nor accelerates according to the local IMU measurements.

Now, let's place one of these telescopes in orbit around a non-spinning mass and another in orbit around an identical spinning mass. If we look carefully through the telescopes we will see that both telescopes exhibit parallax (nearby stars will shift relative to the distant stars). However, in addition to the parallax, the telescope around the spinning mass will also precess slightly (the whole star field will move in a circle). This does not occur with the telescope around the non-spinning mass.

This effect is called frame dragging. It is not predicted by Newtonian gravity but it is predicted by General Relativity, and has been measured (though not to great precision) with Gravity Probe B.

As far as "what is being dragged", the IMU experimentally defines a local inertial reference frame. That is what is being dragged. Local inertial reference frames are dragged so as to change their orientations with respect to distant objects.

I dislike the whole "fabric of spacetime" phrase, so I would never describe something that way and “dragging the gravitational field” is ambiguous since there are multiple things that could qualify as the gravitational field in GR (the metric, the Riemann curvature tensor, the Christoffel symbols). So I would simply describe frame dragging as the dragging of local inertial frames and, if asked what that means, would describe it experimentally as I did above.