I'd like to welcome the opportunity to place emphasis on the Poké Pelago feature that has been around since the vanilla Sun and Moon games. The reason behind this is that nostalgia has recently gotten the better of me and that I decided to revisited the good old short movie "Pikachu's Vacation".
While this feature has undeniably proven itself to be very useful when it comes to leveling up your Pokémon, raising their levels of friendship, harvesting berries and hunting treasures, I'd also like to point out another aspect that I like about it by drawing parallels between this place and the aforementioned movie.
Like Elder Li in Violet City's Bellsprout Tower already adequately formulates: "Remember, Pokémon are not tools" (especially not tools of war) and they do indeed deserve to take a break every now and then after the hard work is done, much like we humans. I think that this greatly underlines how the relationship between a trainer and his Pokémon is equal and bilateral. The two parties both have commitments and responsibilities on one hand, but also liberties and the need for leisure time activities on the other hand.
The Poké Pelago's design is great in that sense because, much like the playgrounds in "Pikachu's Vacation", it has many different versatile isles that appeal to individual Pokémon that have, much like humans, individual needs and leisure time activities. In my mind I like to imagine it being a place where my Pokémon get to know each other better. Some of them may have known each other for a long time, but I personally always like to imagine that this is this the optimal place for newer members of the collection to acclimatize.
I prefer this image to Pokémon just being stuffed into the PC BOXes like you'd stuff working animals and farm animals into a shack or a stable after they've served their purpose.
In my head canon the berry fields are for example quite literally a fertile ground for a "sense of community" and "teamwork" (and berries of course). I like to imagine Grass-type Pokémon developing a penchant for gardening and for harvesting berries, like for example an Exeggutor that happily waters the berry fields with a watering can in each mouth, knowing that it positively contributes to the adventure of its trainer and ergo the rest of the team. I'm saying this because some Pokémon show personal initiative and add fertilizer at regular intervals and some of them also come up to you every now and then whenever you visit the place, in order proudly present you the "fruits of their work".
The same also applies to the treasure hunts. I like to imagine Pokémon like Dugtrio and other Ground-types (most definitely also Rock-types) that enjoy digging in the caves and that donate the treasures that they stumbled across in the process for the good of the community. I like to imagine some of them to look for specific stones in order to help a fellow team member evolve.
As for the training playground, I like to imagine it being a place for competitive Pokémon that like to stay fit and that enjoy practicing with each other (kinda like the Hitmonchan and the Hitmonlee that were exchanging punches and kicks in the hedge maze). Maybe they also give each other high fives compliment each other for pulling off impressive wins together at the Battle Tree for example.
I like to think of my Pokémon as self-aware creatures (yes, even Hidden Ability Clefable and Quagsire) that "pull together" (like in the "Pikachu's Vacation" gif. above) for the sake of the entire team, an image that stands in great contrast to how Pokémon games used to be perceived as a form of promoting mindless cockfights by uninformed and biased "critics" and "experts" back in the day.