Satellites stay up for the same reason a ball you throw doesn't fall to earth immediately after it leaves your hand. For a while, it seems to defy gravity, but friction and gravity work to gradually bring it down. Now imagine you can throw so far your ball makes it halfway around the earth. Now imagine you throw it even harder, so it makes it around multiple times, many many times. Now you have a satellite. For a while, until friction and gravity bring it down. As posted above satellites may have boosters to extend the length of the "throw".
Be careful, though, if you throw faster than 25,000 mph, the ball escapes earth's gravity. If you throw less than 18,000 mph, no orbit. So practice up to get your velocity between 18 and 25 thousand.
I welcome scientific corrections, but I think I explained correctly. Maybe not.