So this is something I've spent a lot of time thinking about, and I've realized that my books don't have the full explanation that I've worked out (and probably never will because it would interrupt the story needlessly for a long, boring lecture). So, for anyone who's curious, here it is.

I'll start by saying that I came up with this on my own after doing as much research as I could. I got some ideas from a theoretical article on Bulbapedia, but I also added quite a lot of my original ideas from watching/reading sci-fi and from being an actual computer programmer.

So my explanation, like any good sci-fi, is based on a little bit of science. I'm talking about the law of conservation of mass-energy, which says that the total amount of mass and energy in a system always stays the same. The law of conservation of mass says that mass can't be created or destroyed. The law of conservation of energy says that energy can't be created or destroyed, but mass can be turned into energy and energy can be turned into mass. The conversion of mass into energy is shown by Einstein's famous formula E = mc^2. M is the mass, and E is the energy.

What this means for Pokémon is that you can't take a big Pokémon with a lot of mass and shrink it down into a mini version with less mass inside a pokéball. That would be destroying mass. The leftover mass that didn't make it into the mini version would have to have been conserved by turning into energy. And then you would need that same amount of energy to be transformed back into mass in order to bring the Pokémon back to its full size form. So that energy would need to be stored.

Hey, as long as we're storing energy, doesn't it make much more sense to store the whole Pokémon as energy? It would be less complicated and also means that we don't have to worry about that mini version of the Pokémon being physically comfortable inside the pokéball.

That's just how it works in my books.  The pokéball works by transforming the physical body of the Pokémon into energy, which it then stores inside wires and circuits, similar to electricity. So the inside of a pokéball is just a bunch of wires for storing energy, the machine that does the energy/matter transformation, and a bunch of scanners and sensors.

One these is a motion sensor, which an empty pokéball uses to tell when it's being thrown at a Pokémon. It then triggers a program to try to capture the Pokémon it touches. It also has scanners all over its surface to get a reading of the thing it touches. It takes a tiny bit of fur or skin and reads the DNA. This is used to confirm that, first of all,  it actually hit a Pokémon. We wouldn't want a pokéball to suck in a person, for example. Secondly, this will be important if the Pokémon is captured, so it stores the DNA pattern.

Next, it starts the process of converting matter to energy, which also involves running a detailed scan of every part of the Pokémon to make a kind of blueprint of what it looks like, what it's body is made of, etc. The blueprint gets stored in memory and the energy gets stored in the circuits. If the capture fails, the pokéball uses the blueprint and the DNA pattern to transform the energy back into exactly the right kinds of molecules arranged in exactly the right places, which demonstrates some pretty dang impressive computational power. Not to mention the tremendous capacity for energy storage, considering that the "c" in E = mc^2 is the speed of light, which is such a gigantic multiplier that I think even "gigantic" doesn't quite cover it. Basically, these things are powerhouses far beyond our own technological abilities.

If the capture is successful, the pokéball stores the energy, the blueprint, and the DNA pattern indefinitely. The next time something touches the pokéball, it also gets a DNA scan. If the DNA is human, the fingerprint scanners on the surface activate. They scan and store the trainer's fingerprints so that trainer only is registered as the owner. It doesn't do a DNA scan every time because fingerprints are just as unique and a fingerprint scan takes a lot less energy. Even we in the real world have fingerprint scanners, so you know it can't be as mega-advanced as on-the-spot DNA scanning.

If a pokéball containing a Pokémon gets thrown, it  uses the motion detector to know that it's just about to be thrown and then uses the fingerprint scanners to make sure the owner is the one throwing it. Then it starts the same process of changing energy back to matter, with just one difference. It uses some of its own energy to create a tiny little bit of matter underneath the Pokémon's skin, sort of like a tracking chip but different. This tiny little bit of matter would be read by any other pokéball that is thrown at the Pokémon, so it acts as a system to prevent theft. A pokéball thrown at a Pokémon that is already owned will read this and automatically not capture the Pokémon. This little bit of matter also lets the pokéball that created it scan and recognize the Pokémon much more easily.

The little bit of extra matter can be removed from a Pokémon by pressing a special button inside a pokéball, as seen in The Girl with the Chikorita. This is useful for releasing a Pokémon back to the wild, freeing it to be captured by someone else.

As noted in Getting Out of Fuchsia, the fingerprint scanners have the added safety feature of not working when a person is unconscious, so thieves can't knock someone out (or kill them, or chop off their hand) to release and steal their Pokemon. They also can't operate another person's pokéball to return a Pokémon that might be attacking them in order to defend its trainer.

Incidentally, returning a Pokémon also uses the fingerprint scanners. It scans to be sure it's the trainer, then waits until the white button is pointing directly at the Pokémon. It scans the Pokémon, takes note of any physical changes, and turns them back into energy again.

All of this means that the Pokémon inside the ball exists only as energy. It doesn't have a body, so it doesn't hear what's going on around the pokéball. It also doesn't have to worry about physical comfort. It doesn't feel cramped. It doesn't need miniature furniture to lounge around on. It doesn't even get bored because it's existing in a completely different way. No one really knows what being energy is like because no one remembers being energy after they come back.

I say "no one" because, in my world, people can be transformed into energy, too. No, they don't go into pokéballs, but they do step onto teleporters, like in Sabrina's gym. I think of these like the transporter in Star Trek. They turn a person into energy, beam them to a different place, and regenerate them again. In different episodes, you see problems with the transporter not putting people back correctly or losing their pattern so they can't be regenerated at all. This sort of thing completely supports my theory. The pattern or blueprint has to be formed completely, stored accurately, and used precisely in order for the Pokémon to be changed back as it should be. There's even an episode of Star Trek where someone gets caught in a transporter for a number of years before finally being released in the exact same physical form, which shows that it actually CAN work like a pokéball.

Just like the person caught in the transporter, Pokémon don't remember anything from when they were in storage. They pop back into existence in a new time and place, and that's the end of it.

The main character of Getting Out of Fuchsia and The Girl with the Chikorita talks about it as a sort of "suspended animation", which is true in that the Pokémon inside doesn't grow or age or change in any way. A baby Pokémon could stay inside a pokéball for twenty years and still come back as a baby with no sense of lost time because the pokéball stored their blueprint as a baby. A Pokémon that gets seriously injured might bleed to death in the real world but would stay alive inside a pokéball because bleeding doesn't happen there. The pokéball saves the Pokémon in its injured state, exactly as it was when the scan took place, and stops it from getting any worse.

Pokémon Center machines then have special programs that change the energy and the blueprint just right so that the Pokémon inside the pokéball is completely healed, reversing any damage instantly. These are run based on a vast database of medical knowledge for every species of Pokemon in existence. Pokémon Center machines also recharge the batteries of the pokéballs, as mentioned in Getting Out of Fuchsia.

Bill's PC (or whoever's in whatever region) is essentially a system for storing extra pokéballs. In the anime, when Ash catches a seventh Pokémon (his Krabby), it disappears right out of his hand. Personally, I think that adding teleportation hardware to the already overloaded pokéball would be a bit of overkill when, instead, a seventh pokéball could just disable itself so that the Pokémon inside is stored but it can't be called back out to give a trainer an unfair advantage. This disappearance would also be less convenient than what I propose.

My proposal is that something has to track how many active pokéballs a trainer has at a given time. A pokéball belt does that by having a sensor in each of its six slots to communicate with pokéballs. Placing a pokéball into the first slot registers that Pokémon as the first of your team and so on for all six. A new capture automatically registers to the next open slot as soon as it gets in range. A seventh pokéball will disable itself upon finding no open slots. This pokéball could be thrown into your bag to be dropped off at a Pokémon Center later. Alternately, you could switch the new pokéball for an old one, overwriting one of the slots. The pokéball you switch it with would then be disabled instead.

Disabled pokéballs can be dropped off at a Pokémon Center for storage. They're placed in a big room in the back, as shown in episode 2 of the anime and in Pokémon Rewritten. The PC computers you see in Pokémon Centers read your fingerprints and query the database for all pokéballs in storage belonging to you. You can then select whichever one you want, and it will be teleported directly to you wherever you are.

Trainers who don't like pokéball belts can use anything else they want to perform the same function. For example, a person with a Pokédex could just stuff their pokéballs into their pocket as long as they manually register their team of six on the Pokédex. The Pokédex then communicates with the pokéballs and serves the same function as a belt.

So that's the basics. Actually, there's even more that I've worked out and some that I'm probably forgetting. If you have questions, feel free to ask, and I'll update this with an explanation for whatever it is. I hope it was somewhat interesting!