At the moment it is Hubble's law and a bit of philosophy.

We observe that galaxies appear to move away from us at a rate that is proportional to their distance from us.

Whilst one could argue that we are at the centre of this very uniform expansion it begs the question as to why Hubble's law should exist. The simplest explanation is that General Relativity applies (as we observe in a number of other cases) and we live in an expanding universe.

In such a universe, the redshift of distant galaxies is not caused by relative motion, but by the expansion of space. At high redshifts, these phenomena become distinct in that the relationship between "velocity" and redshift is different, for instance allowing "faster than light" (apparent) speeds.

So basically at present, expansion fits the facts (far) better than any of the alternatives.

Direct evidence for the expansion is on the horizon though. In an expanding universe, the speed at which galaxies move away from us can change slowly with time (and with distance) by of order 10 cm/s, despite their being no force on them. There are plans to measure this tiny effect with the European Extremely Large Telescope over the course of a decade.