If you've ever saw a professional make pizza bases by throwing them in the air, spinning, it's more or less like that.

More specifically, gravity is attractive in all directions. When the gas cloud the solar system firmed out of started spinning along a specific axis (because of how it was contracting), there was an outward force (the centrifugal force) counteracting gravity in one plane, but nothing perpendicular to that plane. So all the matter started falling into the centre, except where it couldn't, because the centrifugal force kept it in orbit around the forming sun. Repeat the process on smaller scale for planets, and that's why most of them rotate the same way too.