Short answer: you stop nutation the way you stop any oscillation, by giving it an appropriate opposite impulse as it crosses zero, the oscillations central, zero energy point.
Longer answer: The terminology in this area can be messy. So let's start with a simple model.
The classic motions of a spinning top or (Navy) gyroscope are "spin", "precession", and "nutation". Spin is just what it does around its initial axis: You "spin it up". If you stand the top on its point, in will eventually rotate around at an angle due to the torque of gravity trying to make it fall over: That's precession. And at the start, as it falls from vertical to its final precession tilt angle, it'll overshoot and bounce up and down a bit at first: That's nutation. It's an oscillation around a stable precession angle. (The formal definitions that go back to the first, 2nd and 3rd Euler angles)
That up-down oscillation can be avoided by starting the top's motion near it's final angle and precession angular velocity. If the rotation starts away from its stable angle and velocity, you can null the resulting nutation with appropriate torques applied as it goes through that stable point. The stable point could either be calculated, or just approximated as mid-way between the limits.
You sometimes do see true precessional motion of aeronautical craft, but I think it's pretty rare because it requires a constant co-moving torque. Maybe an asymmetry in the aerodynamics, rotating with the craft, could create a torque that would drive precessional motion.
Usually, you end up considering a body that's more complicated than a top. If you've got an irregular body, what's spin, what's precession and what's nutation depends a bit on how you look at it.
For example, say you want the primary spin axis of a craft to be along some line, perhaps straight through the pointy end. If there's a small asymmetry in the mass distribution (i.e. a little more on one side than another), the natural spin axis won't be along that desired one, it'll be displaced a bit to line up with the actual moment of inertia tensor axis. Now a "pure spin around actual axis" will look, depending on details, like spin+precession (called "coning") plus maybe nutation (in which case it's called "wobble") around the intended axis. Ditto for the effect of asymmetric or non-period torques: They can drive precession and nutation.
The recipe for stopping a notation-like motion is the same: Find the center of the oscillation, and provide an impulse against it. I'm not sure how that interceptor decides how many of the little motors to fire, but perhaps it's just got a look-up table based on the amplitude of oscillation it measures.