To expand on the CMBR bit here for readers who aren't as familiar with cosmology: the current temperature of the CMBR is a result of redshift, but that's not what establishes the need for expansion. No, the problem is that the CMBR seems to be in thermal equilibrium (or more properly to have been in thermal equilibrium at the time of its emission a few hundred thousand years after the Big Bang).
The problem with that is that, if the Universe had remained the same size and has the finite age it appears to have, there would not have been time for information from the "north" side of our observable Universe to reach the "south" side. After all, the "north" side's light is only just now reaching us, at the center of our observable Universe, and would have to travel a very long way again to reach the south side. In cosmological/relativity terms, we say they're not causally connected.
The obvious explanation for this is that the early Universe expanded from a very small region that was causally connected for long enough to reach thermal equilibrium, and then expanded. So while the "north" side of the observable Universe and the "south" side of the observable Universe cannot communicate from their current positions (and in an expanding Universe will never be able to do so again), they were able to communicate in the past for long enough to establish equilibrium.
Without expansion, you need a way for two patches of sky that have never been able to communicate with one another to somehow "agree" to be the same temperature. And that's pretty hard to do.