Looking back I definitely appreciate the time and effort my parents put in to our activities. As previous posters have said, the opportunity to share an interest with like-minded peers, to regularly interact with a non-school group of children, and to have the satisfaction of spending time on and improving in areas where I had a genuine interest were hugely valuable.
I started after school drama classes when I got old enough that playing "pretend" without an excuse became socially uncool. After a few years I lost interest, but the skills and confidence I got from drama definitely helped me when I took up debate as an activity in school and university, and having done debate definitely helped me in law school and my legal career.
There wasn't any way that anyone could have known that's how it would work out when I was seven, though. It could have been art class (I quit as soon as I could), swimming (I stuck with it for years, and as someone who always hated school sports it was great for my fitness), after-school science classes from the local museum (I remember "Slimy Things" and "The Science of Crime Solving" with particular fondness, but I was always going to be more of a humanities person), or music lessons (I really liked my teacher and playing in school ensembles, but in retrospect the flute was not the right instrument for me) that stuck with me.
In most cases, I think you'll have a hard time drawing a straight line between childhood activities and measurable success in later life, but in terms of generally enriching children's views of the world and developing "soft skills," they can be fantastic. Obviously, there's a balance to be struck between activities, free play time, and family life, but for a child who is happy and engaged and getting something out of them, I think they're worth the parental effort.