Well, it's hard to know what such statements mean, as they're mostly not very precisely worded. For instance, I'm not sure how to interpret the concept of an 'unstable rider', unless perhaps as a rider for falls (or jumps) off the bike.
In any case, interpreting it as meaning that the rider has an impact on the stability of the machine-rider system, then that is certainly the case and there are many ways in which that happens. One is, as Geesxara notes, that the rider can, to a substantial degree, affect the center of mass of the system. This can then induce load transfer, change the geometry and hence the stability and handling characteristics of the machine. This is the purely static viewpoint (i.e. where we consider what happens after the new configuration has been achieved, after say the rider has shifted violently to left of the center plane of the bike and is now hanging off, the machine having achieved equilibrium once more).
A perhaps equally important aspect, has to do with the dynamic response of the system, where violent shifts of the rider's body, will necessarily be accompanied by reactions on the chassis, unsettling it in many ways, which are too complex for me to talk about, but which should be confined to the transient period during which the rider is moving, unless they destabilize the machine to the point of starting a wobble or weave perhaps.
All the above has assumed changes of the rider's body position, which have not fed back into the steering column. If you shift your body and push against the grips to do so, then it will have obvious destabilizing effects on the machine; obvious, that is, while contemplating them, although they may well be far from obvious while actually causing them. For instance, braking and bracing yourself against the grips, may well feel like the bike is becoming unstable due to an impending wheel lock.
Instead of inadvertently actively forcing the grips (due to reasons such as the above), you might also affect the machine by fighting the forces fed back to the steering column from the road, due to the bike's geometry. For instance the caster geometry of the front wheel has a self-stabilizing effect, which works by pushing the wheel (and hence steering column) into alignment, as soon as it rotates away from the direction of the bike's motion. This is counteracted by such things, as the friction in the head bearings, the steering damper and to some degree your hands, which are generally expected to introduce some damping force. If you maintain a death grip on the grips though, say because you've become rigid due to panic, or because you're keeping your arms straight and support your body on the grips, then you're fighting (or perhaps downright preventing) that stabilizing effect of the front wheel geometry, causing the bike to become more unstable.
This all assumes purely dynamic effects, and ignores the equally important case of plain old bad input, such as chopping the throttle, pushing too hard on the bars while turning, etc. and there are probably other ways, in which a rider can cause a well-engineered modern machine feel like a scooter from the fifties, which are escaping me.