Cubing is a way of calculating a price for shipping an item based on the volume it takes up, rather than the actual weight.
It might sound weird, but if, for example, you were to send 50 bean bag refills, you could fill up a truck. If shipping were only charged on weight alone, then that full truck would have only had charges collected on (say) 50kg - if the bags were 1kg each. When the truck might be capable of carrying 1 tonne, you can see that trip for the truck is going to be running at a big loss. Cubing addresses this problem.
They make it easier to work out a price by using a formula to calculate an equivalent weight (called the cubic weight) and then that is used to look up the normal weight tables.
Although in some parts of the world they use a different figure, the reference used by Australia Post (and a number of freight companies) for cubing is 250kg per cubic metre - which is equivalent to a 10 x 10 x 10 cm package weighing 250g. (3/4 air and 1/4 water, for example)
While every parcel has a volume and, thus, can be cubed - the means of correctly doing this is only straightforward when the parcel is of a simple box shape: Length x Width x Height x 250Kg/m3. Add curves, irregular angles or indefinite edges (such as when shoved inside bags) and this simple formula is no longer usable. Your Australia Post person who pushed the corner in made it non-rectangular and was able to charge you the normal weight based postage - because it could no longer be accurately assessed with the simple cubing formula they have in their computers. While she did you a favour, I don't think the Australia Post hierarchy would be pleased.
Australia Post do not cube (or try to cube) everything - there are threshholds below which they just don't bother. For example, if the destination is within 50km of the sender. My truck example above isn't particularly relevant to short trips around suburbia - but when you start loading interstate trucks and especially planes, you can see it becomes much more of an issue.
If your parcel DOES come under the cubing rule, then they take the 'cubic weight' as described above and the actual weight, then use the greater of these two to look up their rate tables.
One thing to remember is that cubing will only cost you more in postage for parcels that are lighter than the reference figure ... anything heavier than that will be charged on actual weight - just like it has been for decades.