I've just finished reading The Hidden Lives of Trees by Peter Wohlleben, which I thoroughly enjoyed.
According to the author, the advantage of shaking the leaves is that both sides of the leaf are exposed to the sun, and therefore both sides can photosynthesize. This would be in contrast to other species, where the underside is reserved for breathing and transpiring. However, to me (a complete amateur), the underside of a quaking aspen seems paler in colour and doesn't "look" like it's any more suited to photosynthesis than the underside of any other leaf.
In addition, I can't find any other sources that back this claim. Most other sources (for example https://www.nature.com/scitable/knowledge/library/case-study-the-glorious-golden-and-gigantic-13261308/) seem to claim the shaking of the leaves reduces the risks that come with too much sunlight, which damages the leaves and actually lowers the rate of photosynthesis.
My question is therefore, what is the most convincing advantage the quaking aspen gains from this characteristic ?