My understanding is that the sun and the planets are all moving together through the solar system at the same velocity, and there is no friction involved, so there's no dragging or indeed motor force required.
Drag might be a poor word choice in this context. It doesn't refer to friction or air drag. And certainly nothing to do with a motor. I think the author is using drag in the "carry along with it" sense of the word.
What the author might be trying to say is the planets orbit the Sun while the sun moves through the galaxy. So the planets also must move through the galaxy in a way that closely mimics that of the Sun.
Similarly, the author might also be saying that the Sun exerts a force on the planets that causes them to "stay close" to it. I suppose this could be interpreted as correct since if something exerts a force on only the Sun, the planets will likely keep on orbiting it as long as the force isn't too large. But in reality, the gravitational force by the rest of the galaxy has the same effect on the Sun and on the planets, so this effect likely isn't as big as one might expect.
To make this last point more clear, consider our moon's motion in our solar system. It goes around the Earth, which in turn goes around the Sun. (Therefore the moon's path in the solar system has a sort of Spirograph shape.) But if the Earth weren't there, the moon could orbit the Sun just fine without the help of Earth; it just wouldn't have the smaller revolutions due to its motion around the Earth.
I'm sure celestial mechanics are cringing at my terminology. Sorry.