A home stool test kit is exactly as it sounds, a test kit you take home to test your stool(poop) with. It's quite simple & sanitary, so don't worry, you won't be mixing vials of poo with different solutions or anything like that. The reason you'll take it home with you is due to convenience and privacy. You'll need to collect a sample on three different days. This would be inconvenient to have to go to your doctors 3 days in a row. Also I am sure most people are more comfortable going to the bathroom at their own home.
You also don't have to worry about marching the test back to the doctors office. It comes with a biohazard safe mailer that you put it in & then you drop it in the mail.