Sweetsilver and Miranda covered it pretty well.
My story is a bit different because my kids started out at our public school. My oldest was brough home mid-third grade. She was ahead academically, according to school standards, and was bored most the time. We brought her home, but kept her in the gifted pull-out program and never looked back. For us, it was better for her to be at home. By third grade, we already experience catty girls, playground politics, and a desire to be popular. This caused her a great deal of stress and she wasn't very pleasant at home. Within a few months of homeschool, she was enjoyable to be around! We stumbled around a bit, trying to find out how we would make homeschool work, but it was so much better than school. Eventually, we became part of the "eclectic" groups of homeschoolers. She is now entering 9th grade and will take Spanish, Dance, and Drama at the local high school. If she decides that she hates it, she can drop. If she loves it and wants to become a full-time high school student, that is fine too. I really do think that she will remain a part time student though. She values her free time a lot and she is heavily involved in dance -- she is at the studio about 15-20 hr/week and I think having a pile of homework would really cut into that.
My middle was brought home mid-first grade. I suspected she was dyslexic, the school said she wasn't far enough behind to help. She had intense anxiety regarding school. We were seeing a GI specialist because of a bunch of issues. No cause could be found. Within two weeks of pulling her out, ALL of her symptoms disappeared. I truly believe that the anxiety she felt with school was manifesting in her gut. I feel bad for all the tests she went through, but the Dr. didn't think stress could make this big of an impact. Because I thought she was dyslexic, I researched how to teach a child with dyslexia. This is part of why we are eclectic. She had a great desire to learn to read. I didn't want to wait and see if age would 'fix' this. We tried a combination of things -- some curriculum, some games, some therapies to help her read. I wasn't comfortable waiting (as many unschoolers would). At the same time, perhaps they wouldn't wait since my dd had such an interest. Anyways, reading instruction was deliberate. She hates workbooks, but we have one for math. We don't always use it though. We never used any pages about clocks, measurements, or money--that was always done hands on and/or in real life. She loves science so much that I wasn't able to keep enough on hand. One year, she was totally into dissection. I have a science curriculum that I love. I used it as a guide for me to keep her going. When we do dive into something (like dissection) I will put the curriculum away until we are ready for it again.
My youngest was brought home after first grade. She had no problems with public school. She is very smart, but wasn't yet bored. She learned to read with ease. We didn't care for many of the girls in her grade, but that wasn't the reason either. She finally wanted to come home to learn. For her, and all my kids, homeschool is a choice. She decided to come home because she didn't want to be at school all day and then have homework. She was the youngest in our family and the only one with homework. She knew that they older two were done with school around the time she had lunch. She wanted what they had. With her, she loves workbooks. She likes to see her progress. She was this way before she ever started school--so it isn't a product of public school. I am not a big fan of workbooks, but I do oblige. She has a handwriting workbook, a phonics workbook, and a math workbook. The phonics is completely review for her. The math sometimes has to wait until I explain something. I like to use games, real life, and hands on manipulatives to teach the math. It takes her about 30 min to do her work. The rest of the day for her would look more like an unschooler's day. She builds things, cooks/bakes, plays outside, builds forts, reads, recently she has discovered woodworking. She loves to collect bugs and help in the garden. I try to notice her interests and provide nourishment for them. I do have science curriculum, but I haven't used any for her. She is great at exploring and finding things to do/learn about through life.
Regarding socializing. We haven't ever had a problem with that. It is something people worry about until they actually start homeschooling. Then, you realize that homeschooling naturally provides more opportunities to be social than most schools. We don't live in a secluded area, so that makes it easier. My girls participate in sports, dance, girl scouts, drama, park days with other homeschoolers, field trips with other homeschoolers, swim team in the summer, and more. They interact well with people of ALL ages. They have also encountered many kids with differing needs and have learned to do well in those situations too. My oldest also has a steady income because she can babysit during the day.
Mom to three very active girls Anna (15), Kayla (12), Maya (9).