Imagine a ruler balanced in the middle resting on a pencil.
Put a weight on one end and the ruler tips down towards the weight.
Now move the pencil until the ruler is balanced again.
The CoG has moved towards one end - the moment arm (between the pencil and the other end of the ruler) has increased.
If the CoG moves forward the aircraft is becoming nose heavy - as with the weight on the ruler.
So, whilst the moment arm has increased if may not be possible to raise the nose because the elevators don't have enough authority (aerodynamic force) to overcome the effects of the aircraft being nose heavy - due to the CoG being outside its forward limit.
Same goes for being tail heavy when the CoG is outside the rear limit.
The aircraft will be in balance and controllable when the CoG is inside the limits. It will become uncontrollable if the CoG is outside these limits.
Practical examples:
With passengers in the back of a 4-seat light single, it will require nose down trim (up trimmer, down elevator, force 'pushing' up) to counter the rearward movement of the CoG due to the weight of the passengers.
Some aircraft can't carry out aerobatic manoeuvres, such as spinning, if the CoG is too far back, as the aircraft would have a lack of (down) elevator authority to recover from the spin. So there are limitations on weight and balance (no rear passengers, for example) on some aircraft types.