The AD574 datasheet says the following:

How do you calculate that 1.5 Hz is the maximum frequency?

To answer the "why" for this part, it is a very old part; the rev B datasheet was released in 1982, more than 40 years ago. It does not contain a Sample and Hold (S&H) circuit that, as it's described, samples the input then holds it at that value while the ADC figures out what voltage the input is.

Why does that matter? If the input to the ADC is changing, we can't accurately tell what the voltage was when we convert; it will be "smeared" across time. The ADC does not work instantly it takes serval steps to converge to it's final value.

How can we calculate the maximum input frequency that maintains the rated accuracy?

The peak rate of change of a signal is given by:

$$2\pi f A$$

where f is the frequency (in Hz) and A is the amplitude. To get 12-bit accuracy, we need the signal to change by less than 1 bit (first order) during the conversion time.

For this ADC (using the 10V span as in your datasheet snippet), the input is 10V peak-to-peak, so the amplitude is 5V, and the conversion time is 35μs. We need the change in voltage during this time to be less than:

$$\frac{10V}{2^{12}} = 2.44\text{mV}$$

So we can calculate the maximum frequency as:

$$f_{\text{max}} = \frac{2.44\text{ mV}}{2\pi \times 5\text{ V} \times 35\text{ μs}} \approx 2.22\text{ Hz}$$

The 1.5Hz figure given in the Datasheet implies keeping the signal steady to 12.56 Bits.

It describes later in the passage you quoted that you can extend the input signal bandwidth with the same speed by adding an external S&H; these days, they are integrated into almost every ADC on the market.

The datasheet isn't great here. The ADC you specify does not have a sample and hold, and is thus at the mercy of how fast the input data changes.

I'm not going to go through and do the math, but here's my assumption: in order for a full scale sine wave to be properly sampled within 35 microseconds, the input wave must not change by more than 1 LSB in 35 microseconds, and a pure full scale sine wave can meet that criteria if it is below 1.5Hz.

The data sheet goes on to say that if you want to do better, you need to use an external sample and hold, and tells you how to do that.

If you are just sampling? 1/35us = ~28.5kHz but if you want to use the full range of the ADC ±10V and have no error the max signal rate is 1.5Hz.

So you can sample fast, but you can't exceed a certain slew rate if you want error free data.