The tectonic plate boundaries are physical and directly observable. In other words, if you go down in the ocean to a plate boundary, you see giant ridges. But scientists get better details about the structure of the earth using seismic waves, such as those created by earthquakes. Seismic waves bend and bounce as they go through the earth depending on the structure. Mathematically you can extract structure out of many seismic readings, a little like how a CT scan takes several X-ray pictures of the brain at different angles and can then mathematically create a 3D image from these pictures.
The scientists determined the boundaries of the tectonic plates after checking the earth's crust and how they were divided. The theory states that the crust is divided into different sections named after the continent they are located. By plotting the locations, they were able to determine the boundaries and the various plates move.