Have a look at the linked documents giving the popular science background.

They led the teams that gave the decisive proof that there is black hole at the galactic centre. The teams developed techniques to precisely measure the orbits of the stars which orbit the central black hole. Looking at the speed of the stars, particularly as they pass close to the object, you can measure its mass and put a constraint on its size. The only thing which is small enough and heavy enough is a black hole. Doing this required lots of advances in data analysis and adaptive optics to be able to measure the stars orbits to enough precision.