I think the thought is that:
using GDP alone biases towards countries with large populations (like China) which might not have the level of resources implied by a high GDP as their large population drains much of that wealth.
On the other hand, using GDP per capita biases towards very small rich countries, that may not have the power implied by their high GDP/cap, as their size limits how powerful they can be (e.g. Singapore).
So, you instead use a measure in between the two, and the naive way of doing this is by multiplying the two measures. One way of thinking about it is that GDP represents the total resources available to a state, while GDP per capita is a rough measure for how efficiently those resources can be put to use.
Another intuitive way of thinking about this is the surplus domestic product (SDP), which if I recall correctly results in similar rankings as GDP * GDPPC.
Fixed, thanks.