the body-fixed frame

The body-fixed frame is a set of coordinate axes painted onto the rigid body itself, so they rotate and tumble along with it. If you glued a tiny set of x-y-z arrows to a spinning top, those arrows are the body frame. From inside this frame the body never moves -- every atom sits at a fixed coordinate -- while the outside world appears to spin.

The whole point is that in the body frame the mass distribution is constant, so the inertia tensor is a fixed matrix of numbers rather than a time-varying one. Almost always the frame is anchored at the center of mass and aligned with the principal axes, making the inertia tensor diagonal. The catch is that the body frame is non-inertial (it rotates with angular velocity omega), so time derivatives must be converted with the rule (dA/dt)_space = (dA/dt)_body + omega x A for any vector A. That single rule is what turns Newton's rotational law into Euler's equations.

You use the body frame whenever you track a real spinning object -- a satellite, a gyroscope, a molecule. The trade-off is honest bookkeeping: the frame makes the inertia tensor simple but the frame itself is accelerating, so fictitious (centrifugal and Coriolis) effects enter through the omega x A term. To recover the orientation seen by a lab observer you compose the body frame with the space frame using the Euler angles.