I was homeschooled during highschool and I'm familiar with the whole "unschooling" movement. Traditional homeschooling tends to revolve around having a curriculum that mirrors what's taught in public school, but taught in a 1-1 or small group environment by parents. Some of the benefits of homeschooling vs just learning in the regular classroom is that you can usually move at a pace which fits the student (often faster), and avoid disruptions from people who have no interest in learning.
Unschooling was/is a movement within the larger homeschool movement. It's not a new idea and I remember people first talking about it 10+ years ago. The difference in unschooling vs homeschooling is that instead of having classes in particular subjects like math, english, biology, social studies, physics etc. the student directs the studies and these subjects get incorporated into things which interest the student. This can be a very successful way to learn since everything you learn is related in some way to your interests.
For example if you really wanted to design computer games your english/writing work could be incorporated into doing a design doc/writing a story/dialog for your game. Your math/science work would be incorporated into the project as well since you'll probably want to know physics/chemistry so you can make your game realistic, and you'll probably need math all over the place to design various game systems. The big trick with unschooling is that you and/or your parents have to see how to incorporate your interests into everything you're learning.
On unschooling story I remember was someone who wanted to breed championship race horses so they learned a ton of biology, enough math to manage a farm, and spent lots of time hanging out with their horses. Personally the few people I knew who did unschooling where really into theater and so focused their studies around acting, and skills which would help them get a career in theater. So rather than have someone take an math class with just "boring" numbers they'd have to learn about the costs of putting on a theater production, balance books, selling tickets, figuring out dimensions to build sets, and other theater related math areas. Sure these people didn't do calculus, but then again most people don't do calc in highschool (not needed unless your into science). Basically, the unschooling philosophy was let you kid do what they want and drive them toward learning what their passionate about. This is very similar to the environment one encounters in graduate school while doing Masters or PhD research. But not everyone can succeed in such a free learning environment.
The challenge with unschooling is that it only really works with a special sort of person who's HIGHLY motivated. These people are the types who tend to succeed wherever they are. If they weren't unschooled they'd probably be doing this stuff in their spare time (possibly flunky some classes they found "boring"). For every person I met who was really unschooling and learning real skills which would help them out, there was someone else who was unschooled because that sounded better than "staying home and playing video games". I always felt when I met an unschooler/homeschooler who's parents didn't push them or weren't encouraging learning did a big disservice to their kid. But I also met many people who learned a lot outside the regular school system and turned out to be productive members of society. The people who don't push and help their students thrive and claim they are unschooling by having their kids play video games all day just give a bad name to all the people who really benefit from such an open learning environment.