real answer: it uses three facts:

- the arbirtrary movement of a rigid can be seen as a translation through any of it's points $P$, whose image is $P'$, and a rotation by some axis passing through $P'$. **Valid to any point $P$ on the body**. (Chasles theorem)

- the center of mass of a rigid body can be seen as a point of the body (with mass zero, only kinematically), meaning it's distance from the rest of points remain unchanged (easily checkable with previous fact and definition of COM).

- If a body rigid movement (continuous in time) is such that it keeps one of its points in the same position, then it must be equivalent to a rotation around some axis that passes through the fixed point.

So here it is: an infinitesimal movement of a rigid body can be understood as a translation and a rotation through (an instantaneous axis that passes by) the center of mass. If the net external force is zero, than the total acceleration of the COM is zero (famous result), but, as the center of mass in on the instanteneous rotation axis, it only has translational acceleration. Thus, the translation acceleration of the system is zero with respect to the COM. If the net external force is not zero, this argument guarantees that the COM will transladate.

If the net force is zero, we know that the center of mass is not moving (accelerating). From facts 2 and 3, the system will be then rotating about some axis that passes through the COM, and thus the system as a whole must have ZERO translational acceleration.

Thus it is necessary and sufficient that the net external force is null in order for a rigid body to not transladate.