Why do astronomers theorize that most of the matter in galaxies and clusters of galaxies is dark matter?
Something must be exerting gravity to hold the stars in orbit around the center of the galaxy, and the known mass of the stars is not enough (even with a large black hole thrown in at the center).
The Observational Evidence section of this article goes into more detail. Without dark matter, stars revolving around the galaxy's center should be moving fast enough to escape, blowing the galaxy apart.
Astronomers also see light being bent by the gravitational field of other galaxies, and that measurement too shows evidence of "dark" mass. There even appear to be filaments of dark matter between pairs of galaxies.
The blue ring in this image indicates that there is Gravitational Lensing that is done by dark matter. The ring itself actually is dark matter that is doing the Gravitational Lensing.