Short answer?
The "how" ranges from extreme daily physical workouts in the gym + dance practices to the last worst choice of those horrible, insane 'diets'
(we're talking some fruit for 'breakfast,' maybe a sweet potato for 'lunch,' and maybe some cheese and an apple for 'dinner.'

Sure, there are some who manage to restrict their food intake and still remain relatively healthy, but overall there is far too much of an emphasis in SK on looking pale and thin - which isn't good, especially for a young person anywhere between the ages of 17 to 30. The damage this obsession with thinness is doing to their young bodies is terrible - many people who know the torture involved in staying slim like so many idols do are fully expecting Gangnam in 20 to 30 years to be full of relatively thin former idols who look far older than their actual age and either can't walk at all and are in motorized wheelchair scooters because their crazy diets have ruined their joints, or are walking slowly and painfully with canes or walkers. It's a nightmarish scenario, but it's really possible.

tl;dr - You really, really DON'T want to be as thin as K-pop idols, really. You don't.

"How do they stay so skinny?"

Simple: by not living a balanced life when it comes to nutrition. They eat very little, they starve themselves, their weights are controlled by their companies, they are weighed regularly, they go on unhealthy, unsustainable crash diets ahead of comebacks, the expectations on their bodies are very strict, they are praised for being skinny as stick figures and try to live up to those compliments, they don't care if they lose muscle mass, they choose losing muscle mass to stay super-thin in favour of having a healthy amount of muscle mass, they exercise obsessively when it's time to meet their target weight, they eat in secret to not get yelled at by their companies and often feel guilty about it, hunger is a constant feeling for them, and many of them go through periods of eating disorders, some even have to take breaks from K-pop or leave all together because of that.

So, good luck with that. And "how many calories per day?" That's obviously an individual issue. One can't determine one's own dietary plans and calorie targets based on others'. You want to be slim? There are many ways to achieve that healthily and sustainably, a K-pop lifestyle is not one of them.