When they were tiny, to get me out of the house and fend off depression.
A bit bigger, to improve their physical and social skills,
Even bigger, because they actively enjoy said activities, meet a whole new group of friends for each of them, and are learning to be community minded citizens who care about other people.
Two out of the the three of mine have specific disabilities or additional needs which mean that hiding at home isn't a great choice for physical or mental well-being. Dd2 does a number of sports and dance because it reduces her physiotherapy burden (call it community-minded of me to pay for it so that you don't have to) and a few others because she needs to be used to meeting people and getting on with life independently. It also means that despite her disability, she gets to take part in competitive sports without knowing she will always come last...
Ds would hide in the basement and play video games otherwise (ADHD with ASD tendencies). Dance and swimming give him the opportunity to exercise (ADHD bit) and the Magic-The Gathering clubs mean that he gets to actually socialise with like minded nerds (the ASD bit).
Dd1 (the only NT kid in the portfolio) dances, skis, swims. She keeps very fit, and the dance is brilliant for enforcing the discipline that's required to maintain her school grades. It also rather handily has provided her with a means to earning money as a stand-in dance teacher, and she's on her way to life guarding.
The girls are both in guiding, as it provides them with all female spaces should they need that, and instils leadership and independence. They also choose to work for the various awards in said organisation, which teaches them goal setting and instils a work ethic to reach said goal. It has also provided dd1 with her emergency first aid cert, which is a bonus for babysitting gigs...
Ds left scouts (his choice).
Honestly, I'd keep them locked up and save my money if I could, it costs me a fortune. But so far we have moved very frequently (every year or two) and extra -curricular are a fantastic way to meet new friends when you are uprooted periodically. They are thriving. I don't force them to leave the house with a pitchfork. I would rather they interact with other people and learn more about their communities, and the opportunities therein. I don't prize a hermit like existence, although I would enjoy it myself... 
I'm happy with what they do, they are happy with what they do. I am happy that other families do nothing and stay home. Horses for courses. But extra-curriculars have kept this family both mentally and physically healthy through all sorts of upheavals, both literal and metaphorical.