I find that when I make stews, casseroles, curry’s etc. that they are much nicer after a day or two in the fridge. I understand that it “gives the flavours time to meld” etc, but what actually happens? Is there a way to make it happen first time round?
I think that your thinking in terms of "leftovers" is what is tripping you up here. Cooking food involves tons of chemical changes, and some of them happen best on the stove, and others need a long period of resting time.
If you have a recipe which tastes better on the second day, then all you have to do is to plan to serve it on the second day. See it as a recipe whose last stage needs 24 hours and happens at fridge tempearature. You wouldn't serve yogurt before it has spent 8 hours at 46 C, because the mixture doesn't taste like yogurt yet. Similarly, you shouldn't serve your "perfect beef stew" before it has spent 24 hours at 4 C, because it doesn't taste like "perfect beef stew" yet.
If you are looking for ways to speed up the process, I don't think these exist. Maturing of foods involves complex chemical reactions, which need their time. This is why many luxury food items let you know how long they were aged.
For a bit of background, see also this question on the process of flavors "marrying".
I don't know if it's truly "letting flavors meld", and how much of it's other stuffs going on (like the breakdown of larger carbohydrates mentioned at Serious Eats), and how much of it is change in texture due to cooling and then reheating. Alton Brown made use of this in his Good Eats beef stew recipe by roasting the meat then allowing it to cool.
For the "flavor melding" aspect, you can try to add alcohol (a solvent) to your dish to try to extract flavors faster and let them mix with the rest of the dish more easily. But I would recommend using it sparingly, or the flavor of the alcohol (even for a neutral spirit) can be off-putting to some people.