Objects that are further away accelerate faster than objects that are closer. Quite an odd anomaly for objects moving on a static background. Furthermore, everything in the universe is moving away from everything else dependent on your reference frame. Say I travel 1 light year into space, now I take an observational survey of all of the "fixed" points of light in the space around me. I soon see some of them receding away dependent on my distance from them so it changes with respect to my observations on Earth (Hubble's law). Also, metric expansion only occurs globally, or between galaxies that aren't locally bound. For instance, there is no metric expansion between the Earth and our local cluster of galaxies.
Beyond this, there is the theoretical background which comes from the interpretation of Einstein's Field Equations. Friedmann derived a set of equations which described the non-static universe in Einstein's Field Equations as a change in metric over time.