I don't understand all the chemistry. Here are a few factors. First, gasoline is a wild mix of all manner of chemicals. In the old days, it contained lots of benzene (thence the name!), which is nastily carcinogenic, but today that's removed and regulated. Instead, it contains mostly branched alkanes (lots of iso-octane), olefins (more carbon, less hydrogen, gives it a higher octane rating), and strangely also contains a significant amount of butane. The butane is a gas, and is dissolved in the blend. If you let it sit for a while, the butane and other light molecules evaporate out, making the remainder denser and less flammable.
The next thing is obviously oxydation: The air in the tank contains oxygen, which will turn some of the non-oxygenated molecules into alcohols / ketones / ethers / phenols / all that organic chemistry mess. Again, they don't burn as well, since they are already partially "burned".
Then add ethyl alcohol (or MTBE) to it. Today that's required by law for on-road gasoline in the US. The problem with ethanol is that it is highly hygroscopic, so it attracts water from the air. That's already a bad thing, since water is not burnable, and makes the gasoline denser and contain less energy. But what's much worse is that the water in some fashion (which I do not understand) catalyzes reactions where some compounds in the gasoline polymerize, meaning create large molecules that are insoluble. I think that's the stuff that makes old gasoline become dark yellow, and can form large plastic-like clumps. Sometimes when cleaning a carburetor that had old gas in it, you can find large deposits which are like ear wax.
So when you have old gasoline, mixing it with new one is not a good idea: The attracted moisture will now proceed to ruin the new gasoline. The best thing to do is to get rid of the old gasoline, for example by burning it in an engine! If I have some moderately old gasoline which I don't want to run through small engines (like chainsaws or trimmers, or little 4-cycle Hondas), I put a gallon or two of it in my big V8 pickup truck, which already has 20 gallons of fresh gasoline. With modern engine controls, the slight drop in octane can easily be handled by the big engine. In my opinion, the trick is to add a small amount of stale gasoline to a large amount of fresh stuff, in a vehicle that will get rid of it quickly.
Obviously, another thing to do is to prevent aging in the first place. One thing to do is to add things like Stabil to the gasoline. Another option is to buy "racing gasoline" which is alcohol free, because it lasts much longer (but is also heinously expensive, and you have to be careful to not buy leaded gasoline by mistake).
