Ethan Siegel is referring to a problem in cosmology called the horizon problem.
If you look at any object and you find it has the same temperature everywhere then you assume this is because the object has had the time to settle down to a uniform temperature. For example if you start with a tank of water that has hot water at one end and cold water at the other then if you simply wait the hot water will cool down and the cold water will warm up, and the water will end up with a uniform (lukewarm) temperature.
If we look at the universe around us then we see it has the same temperature everywhere. We measure this temperature by measuring the cosmic microwave background, and it turns out to be about 2.7 Kelvin. This would be quite reasonable if the universe has had time for the temperature to even out i.e. for any hot areas to cool and any cool areas to warm. The trouble is that we can use general relativity to work out how the universe expanded from the Big Bang, and if we do this we find that the universe didn't have enough time for the temperature to even out. It expanded too fast for any hot areas to heat up the cool areas or for any cool areas to cool down the hot areas.
So the problem is that we see the universe has the same temperature everywhere, but since it didn't get time after the Big Bang for temperature variations to even out. This means something special must have happened at or after the Big Bang to make the temperatures the same everywhere. And in the conventional Big Bang theory there is no such effect, so the conventional Big Bang theory has no way of explaining why the temperature is so even.
Now, to get back to your question. You ask:
Does big bang predict the size of the fluctuations in the CMB
And the answer is that the conventional Big Bang theory cannot predict the size of the fluctuations in the CMB. All it can say is that there could be arbitrarily large fluctuations because there is no mechanism to smooth out the fluctuation.
So Siegel's statement is phrased badly as it does look as if he is implying the conventional Big Bang theory predicts large fluctuations, when what it really says is that it can't predict the size of fluctuations.
But we can forgive Siegel because what he's working up to is pointing out that the theory of inflation does predict the size of fluctuations, and it predicts a size that matches what we see. His point is that since inflation predicts fluctuations that match observation and BB theory can't predict anything about the fluctuations then that strongly suggests inflation is the better theory.
For completeness I should point out that there is a lot of scepticism about inflation for various technical reasons, but that's a discussion for another day.
The big bang model does not predict the size of the CMB fluctuations - rather, the universe we see today is a consequence of those fluctuations since all the structures we see today (clusters of galaxies and superclusters, filaments etc) must have grown from structures that are traced by those fluctuations.
Given that, there was an expectation for the size of the fluctuations before they were measured. One can calculate what size of fluctuations can evolve to produce the present day universe, providing you add the right ingredients (including lots of dark matter, which allows the initial fluctuations to be as small as they are).
The fluctuations themselves arise in two ways. There is a component produced by quantum physics in the initial moments. It is thought that these fluctuations are then amplified in scale by a burst of cosmic inflation in the first fraction of a second. It is this process that predicts the (observable) universe should have a very uniform temperature, since all parts of it were in "thermal contact" in the past. The inflation also predicts a rather flat spectrum to the remaining ripples, with similar power on all spatial scales.
A second component is produced by compressions and rarefactions in the primordial gas as the CMB is formed, about 350,000 years after the big bang. These modify the primordial fluctuations in ways that are characteristic of the cosmological parameters (the Hubble parameter, the matter density in the universe, the amount of dark matter and energy and so on). A measurement of the spatial spectrum of fluctuations therefore enables the inference of the cosmological parameters. Or, you could say that the size and spectrum of the fluctuations ought to be consistent with the cosmological parameters that have been inferred by other methods.
