We have no direct observational evidence that black holes exist. We only have been able to measure the mass enclosed in a specific area. For a handful of cases, there are no viable explanations for what objects could be so dense except a black hole. From the handful of galaxies where we infer a black hole, we then make the leap that these other galaxies we have measured contain black holes as well. Because nearly every galaxy that scientists have studied in fine detail requires a central black hole, we infer that the remaining hundreds of billions of galaxies also contain one.
Astronomy magazine subscribers can read the full answer for free. Just make sure you're registered with the website.