The Pokémon series usually sticks to having plots and storylines localized to each of their games. These storylines have often been critiqued as being fairly similar from game to game. The formula consists of an evil team that steals people's Pokémon, with a leader who wants to create a new world by using the game's legendary. However, since Generation 4, Game Freak has been using this formula to plant the seeds to introduce an alternate dimension in the Pokémon universe.
Normally, alternate dimensions find their place in fan-theories and speculation, which the Pokémon series has plenty of. While there is a decent amount of speculation, the existence of an alternate dimension in the Pokémon universe is completely canon as of Alpha Saphire and Omega Ruby, with several obscure bits of Pokémon knowledge backing it up. In fact, Game Freak has been hinting towards this alternate dimension storyline since Pokémon Platinum.
Platinum's story is fairly standard for Pokémon games. Team Galatic's leader, Cyrus, is trying to use the three legendary Lake Spirit Pokémon and two ancient orbs to summon the Pokémon that created Time and Space, Dialga and Palkia. Cryus wants to use these two to create his own dimension.
An important part of Generation 4 is the Sinnoh Creation Myth that players can read in the Canalave Library and Celestic Town. In short, the myth tells a story of how the world was created, with a Pokémon known as the "Original One" creating Dialga, Palkia and Giratina. According to the myth, Dialga and Palkia created time and space, the Lake Guardians created spirits and Giratina created matter.
Cyrus's plan does not work because he does not know Giratina's importance to Sinnoh's creation myth. However, later games may introduce another fault in Cyrus' plan, as Pokémon X and Y introduced the importance of spirits. An ancient king introduced in those games, AZ, attempts to use the souls of Pokémon to create an "ultimate weapon." The amount of energy that AZ's efforts create damages the fabric of space so much that, according to Pokémon Alpha Sapphire and Omega Ruby, an entire alternate dimension was forced into existence.
The fact that the souls of Pokémon are powerful enough to damage space itself is an important detail that connects back to the Sinnoh Creation Myth, as the "Original One" is said to be a physical manifestation of the souls of every single Pokémon. This "Original One" is none other than Arceus, another Generation 4 legendary that created Dialga, Palkia and Giratina. This means Pokémons' souls are actually just a small fragment of the Pokémon universe's equivalent of a god.
As a quick recap, an ancient king known as AZ caused a second dimension to be created using the souls of Pokémon, which are fragments of Arceus' power. However, one of the things that there isn't an apparent answer for is what exactly the alternate dimension is, or if it's appeared in a Pokémon game. There are two main theories for what exactly the alternate dimension is: Ultra Space, from Ultra Sun and Ultra Moon, or the timeline depicted in the Pokémon remakes.
The latter theory has less supporting it. Due to how remakes such as Leaf Green and SoulSilver change and update those games' storylines, it would make sense that these remakes take place in their own dimension whilst the original games happen in a "prime" dimension, so to speak. The main bit of evidence supporting that this second dimension is the remake timelines is that Looker, an international police operative first introduced in Pokémon Platinum, appears in Alpha Sapphire and Omega Ruby.
Looker appearing in these Generation 3 remakes isn't inherently proof. However, he appears with a severe case of amnesia. The last time he appeared in a Pokémon game was when he began investigating the Ultra Wormholes in Ultra Sun and Ultra Moon. It's said that traveling through these wormholes takes someone to a new dimension, giving them a nasty case of amnesia in the process. In short, this theory suggests that Looker traveled from the main series of games to the alternate "remake" dimension.
The second main theory as to what the alternate dimension is that it's simply the Ultra Space presented in Sun and Moon. The main connection to Ultra Space comes from Sun and Moon's Aether Foundation. The Aether Foundation focuses on researching Ultra Space and seemingly theorizes that Arecus is somehow integral to closing the Ultra Wormholes that lead to Ultra Space. Not only do they have a copy of the Sinnoh Creation myth on their computers, but they've also created an artificial Arceus through Type:Null.
The Aether Foundation assumedly created Type:Null to explore this connection between Arceus and Ultra Space closer. It seems that there's a very clear connection being made between Ultra Space and Arceus, which may add evidence to the idea that Arecus created Ultra Space. However, at this point, that's as far as this storyline goes. Sword and Shield's Crowned Tundra DLC does feature an Ultra Wormhole, which is used to explain why countless legendary Pokémon can be found in the area.
It appears that Game Freak is building towards a story of two dimensions crossing over, with investigations being done by key in-universe groups such as the Aether Foundation and the Galar region's Macro Cosmos. This could perhaps conclude in the same region that it began in, as rumors of a Diamond and Pearl remake have recently spiked. Regardless of Game Freak's direction, it's super interesting to see how Game Freak has been building the lore behind this secret second dimension ever since Pokémon Diamond and Pearl.