This is a mini-game akin to Pokémon-Amie or Pokémon Refresh from previous generations. It will become available after you leave Wedgehurst Train Station for the Wild Area.
Setting Up Camp
Pokémon Camp can be accessed via the main menu (by pressing the X Button while exploring). For the most part, you can access it anywhere, but there are some situations where the option will be greyed out, such as when you're traveling across water.
Once you've set up camp, the Pokémon from your party will wander around carefree. You cannot move, yourself, but you can move the camera with the analogue sticks. Other controls include:
Call Pokémon via the A Button: Use this to attract the attention of a Pokémon you're directly facing. If the Pokémon is right in front of you, this will change to Speak instead, allowing you to "converse" with your Pokémon.
Menu via the X Button: This opens up the camp menu, where you can change the toy you're using, start cooking or exit the camp. If you need to leave the camp in a jiffy, you can also press the B Button without needing to open this menu.
Take Out Toy via the Y Button: Choose this to grab a toy from your pockets. By default, you'll have a Poké Toy and a Poké Ball. After grabbing a toy, you can use the A Button to use the toy; nearby Pokémon will react accordingly.
Zoom In via the L or R Button: This lets you zoom in the camera, if you want a better look at your Pokémon or surroundings. While zoomed in, you can still move the camera and use all of the controls above.
Playing with your Pokémon isn't just for fun--your Pokémon will gain friendship and experience points (after exiting the camp) depending on how much you play with them. Friendship is required for evolving certain Pokémon, like Eevee into Espeon, Umbreon or Sylveon. Additionally, at higher levels, it can provide random boosts during battles (only for story battles).
Friendship = Affection
In the past, Pokémon had separate Friendship and Affection stats. The first one is affected by battling with your Pokémon, walking around with it or feeding it vitamins, etc. Meanwhile, the second one is increased by playing with your Pokémon in Pokémon-Amie or Refresh.
However, in Sword and Shield, these two stats have been unified. Additionally, the combined Friendship stat has a limit that you can reach without Pokémon Camp. Conveniently, you don't need to exceed this limit to evolve Pokémon that require a certain amount of Friendship.
In the Galar region, people love to cook curries. Cooking can be done via the camp menu (by pressing the X Button). The following options are available:
Start cooking: All good to go? Then pick this!
Check ingredients: If you're unsure, select this to see what ingredients you have at your disposal.
Check Curry Dex: Yes, there is an encyclopedia for curries. If you speak to the Camping King in the Wild Area, you can let him evaluate your Curry Dex. You'll get rewards for your Curry Dex progress.
Quit: Not hungry or not enough ingredients? You can always cook another time.
Before you get into the meat of things, you'll need to select an ingredient. You can purchase these from the two Ingredient Sellers in the Wild Area or receive them as gifts from Poké Jobs. The ingredient you pick will determine what kind of curry you'll cook. Alternatively, you may press the + or - Button to skip this step.
Next, you can select up to 10 berries. These can be obtained from the various berry trees. There are two factors to consider. Firstly, the rarer the berries, the higher the quality of the curry. Secondly, the flavour of the berries with influence the flavour of the curry (ranging from spicy to sour, or no special flavour).
When it comes to the actual cooking, there are three steps:
Fanning the flames: Wave the Joy-Con (either of them works) up and down or repeatedly press the A Button. If you get a good rhythm going, you should see small twinkling gold stars appear above the curry pot.
Stirring the pot: Move the Joy-Con in a circular motion or rotate the left/right analogue stick. You don't want to go too fast (you'll hear the curry spilling) or too slow. Again, you'll see gold stars if the speed is just right.
Putting your heart in: For the finishing touches, thrust the Joy-Con forward or press the A Button when the shrinking circle is inside the highlighted area. If you can get it inside the inner-most area, awesome. But be mindful of the delay between when you begin the motion and when it actually reaches the pot.
Once you've made the curry, you'll get a score depending on the berries you used and how well you cooked the curry. This will affect how much experience points and friendship your Pokémon gain, as well as the amount of HP restored. Also, if your Pokémon like or dislike the flavour, the experience and friendship will increase or decrease.