The idea that our planet's continents drift around the globe, periodically glomming together and breaking a part, is at least 200 years old,but most geologists didn't believe it until the 1960's, when mounting evidence made it clear that the Earth's crust is broken up into fragment,and that those fragment ,called tectonic plate,are moving. And these days we directly track that motion - with millimeter precision from space.
The common,simplified explanation for why tectonic plate are moving is that they're carried along currents in the upper mantle,the slowly flowing layer of rock just below Earth's crust.