Heh, the lengths some people will got to. That probably isn't the best thing to show someone wanting to know what it's all about.
It's about flavours blending chemically and maturing. It's a very complex subject which any of us can only hope to dabble in and hope for good results. The reason is that we have no control over the actual reactions taking place, and can't know exactly what happened even when it works well.
When you mix a new juice you'll often find the colour is quite light. Over time (days or weeks) the colour usually goes a little darker and the taste matures.
The main ingredient is time, however many will try to accelerate things a little using agitation, heat, and exposure to the air. The mechanisms are complex and probably different for different flavours in different combinations but probably involve flavour molecules reacting with each other to make more complex molecules. Somtimes this will hide a flavour, and sometimes it will change or increase it.
Agitation probably does two things - it might activate the molecules and increase the rate of reaction but mostly I suspect it forces the air in the bottle into suspension, giving the comounds more access to the oxygen in it. This might increase the rate of oxidation and some of the complex chemicals involved may even require this process. Different batches maturing at different rates and giving different results may just come down to how much air was in the bottle. There was talk about larger bottles (100ml) of Boba's Bounty tasting different to the 30ml bottles. This might be due to the fact that there would be a smaller percentage of air in the 100ml bottle compared to the juice/air ratio in the 30ml bottle. One fact that seems to indicate that oxidation is occuring in this process is that mixed liquids will go darker with time. This is often an indicator that oxidation is occuring.
Exposure to heat (ie a hot water bath that the bottle are placed in) probably speeds up the process by increasing the rate of chemical reactions. There's probably good scientific reason to think that heating the juice will make it mature faster, and in fact this can be used in wine production where it is required for the wine to reach maturation faster (not for boutique or classic wines however!). Even well made wines will undergo a pastuerisation process (to kill the yeast) these days, and the manufacturers may not be keen on telling you that this might be force maturing the wine also.
Exposure to the air probably just allows fresh oxygen to contact the liquid and increase the rate of oxidation, same as for agitition. Common sense tells us that if we are agitating to force air into suspension, then letting the juice breathe to replace the oxygen in the bottle is probably a good idea as well.
Which brings us to the fan. Personally, I believe it's probably largely an oxidation process, so while the fan with work to keep the air in suspension it doesn't allow for replacement of the air in the bottle for fresh.
A last word about oxidation and flavourings:
Oxidation isn't always a good thing. There are many things in nature that taste worse when oxidised (bananas, apples, potatoes etc) and others that are improved (can't think of examples but I know they exist). Since were're not chemists we can't really know what is going to happen when we steep (or 'age' if you want the more typical term used for potables) except to try by trial and error and see what happens.
I just mixed up a batch of a smokey cherry cigar that I'm working on. The colour was very light, about like a light honey colour and stayed like that all afternoon. I hadn't agitated it at all. A few hours ago I gave it a good airing out and shake up, and the reaction was almost immediate - the colour went about two shades darker in about half an hour. I'm going to compare it to a batch I made made up ealier and see if it's showing the same changes in taste as well.