Cosmological redshift - How do we know it's not caused by the observer's time dilation?
The reason we think that the cosmological redshift is caused by the metric expansion of space is 1) that there is a well-known, physical mechanism that can cause this effect, and 2) that this mechanism is a prediction of a well-established and thoroughly tested theory, namely the theory of general relativity.

The physical mechanism in question can be realized by considering the propagation of a light ray traveling along a null geodesic in the Friedmann-Robertson-Walker metric describing the expanding space (see e.g. Watson 2000 for a derivation). It is a mathematical result, but it is founded on physics, and verified through experimentation.

On the other hand, there is no known mechanism that would cause time to speed up as the Universe evolves. Moreover, even if there were, this would only make photons arrive less often in our reference frame (compared to photons arriving from a local object), but it wouldn't make the individual photons have less energy (unless they somehow kept oscillating at their original frequency, even if entering a reference frame where time goes faster).

But in a sense you're right that our time goes faster than the time of distant objects: In our reference frame, their time is slower by a factor $(1+z)$, where $z$ is the observed redshift. For instance, the brightness of a supernova observed at $z=1$ declines twice as slow as a "local", $z=0$ supernova. This effect is well-known and "in addition" to the redshift, in the sense that the flux received on Earth is diluted by a $(1=z)$ factor from the redshifting and by a $(1=z)$ factor due photons arriving at a slower rate.