We have no independent evidence for this other than on the cosmological scale. General relativity has succeeded in predicting/anticipating a number of crucial phenomena over the years: The correct amount of the deflection of light by the Sun; The precession of the perihelion of Mercury; The gravitational red shift of light. Astronomical observations have also confirmed, through studies of the Hulse-Taylor Pulsar that gravitational radiation exists as predicted by general relativity. Also, the phenomenon of gravitational lensing predicted by general relativity agrees with the images of an increasing number of distant galaxies.
No one has ever seen space stretch in the way predicted by general relativity. This is not a phenomenon that can ever be under human control in a laboratory. However, if we tried to use other explanations for the red shifts we see from distant objects, you would end up with measurable consequences that are even harder to accept than the dilation of space. Example, if the redshifts were due to the enormous masses of the emitting bodies as predicted by the 'gravitational red shift' in general relativity, every distant galaxy would have to be 100s to 1000s of times more massive than our Milky Way. Objects this massive could cause nearby galaxies to have much larger typical velocities than is consistent with their Doppler shifts. Because the typical speed of a body depends on the square root of the mass of the gravitating body nearby, the speeds we measure for the companion galaxies would be 10 to 100 times larger than what we measure. So, the red shift we see in distant quasars cannot be a gravitational red shift.
This leaves the second mechanism predicted by general relativity as the only one consistent both with the character of the light we see, and the speeds of the distant galaxies in their respective clusters. Distant clusters and galaxies look very much like the ones we see nearby, except that the intervening space is dilating with the Big Bang to produce the red shift.
Also, the dilation of space in general relativity leads to other predictions about the universe and its evolution which are borne out by other observations such as the character and existence of the cosmic background radiation. So, for direct evidence we have the redshifts of distant galaxies, and indirectly we have many of the other features of Big Bang cosmology.