Sometimes people in the audio world use "frame" to mean "set of PCM samples per channel". So a typical stereo audio signal at 44100hz would be 44100 "frames" per second, even though it's really 88200 samples per second. The author seems to be using it that way, but it's confusing because he's also talking about sliding a FFT window over by frames.
> If the raw data is just measures of amplitude at a set frequency
LPCM is literally amplitude samples at a rate. Thinking of the sample rate as a 'set frequency' will lead to confusion (even though it obviously is a frequency). When you're thinking of samples as sequential amplitudes, you're thinking in the "time domain". For oscillations of the signal, that's the called the "frequency domain". Fourier transform is how you convert from time domain to frequency domain.
> don't you have to pick an arbitrary time slice to examine (and so risk losing lower-frequency sounds)?
You need at least 2 samples to make an audible frequency. If you only had 1 sample, you wouldn't hear anything, because nothing would be moving. So at 44100hz of sampling frequency, you can capture 0hz to 22050hz of audio frequency. That's called the Nyquist frequency, and it's always half of the sample rate.