It is said to wait sometime before pressing down the French press, what exactly is the driving idea behind waiting before pressing down the piston?
The process of brewing coffee is chemically speaking an extraction - flavor compounds, caffeine etc. migrate into the hot water (compare to steeping tea). The speed of that process is determined by various factors, temperature and surface/contact area amongst them.
For French press, you use a comparatively coarse grind which translates to less surface area than pour over (finer grind) or espresso (really fine). So you have to give the water-coffee mix more time for a good extraction. The customary time, which on average should give the desired results, is four minutes, but you are of course free to experiment with other timings, especially if you are also playing around with grind size.