A large part of the basis of this question may be due to the student who didn’t pay attention in the history section of the NMR course (and not during the technical details bit either) but I find the question intriguing nonetheless.
My basic knowledge of the history of NMR spectroscopy is that in the very old, almost prehistoric days, Bloch and Purcell performed some experiments that applied radiofrequency to a sample and by chance got some radiofrequency signal returned. Then I have a big black box and arrive at modern spectrometers that tune themselves into hydrogen spectra, carbon spectra and see beautiful Fourier-transformed spectra on my PC screen (unless it’s carbon, in which case the baseline is, of course, fuzzy).
I imagine that there had to be a lot of research invested until it was realised that each signal in the Fourier-transformed spectrum corresponds to a hydrogen atom, that they are shifted around depending on how electron-rich or -poor they are, how couplings work and so on. I can imagine that most of this stemmed from trial and error: a certain combination of magnetic field and radiofrequency gives a signal, Fourier transformation seems to happen anywhere, and once you have rationalised that ‘signals’ correspond to ‘hydrogens’ in a certain type of spectrum (likely the one that gave most resonance anyway) you’re half done and the rest is a walk in the park.
But there is one step that seem very unintuitive:
Realising that ‘signals’ correspond to hydrogens.
There is another important step, namely figuring out that solutes can be analysed if hydrogen-free ($\ce{CCl4}$) or at least protium free (deuterated) solvents are used. Now either it was figured out first that hydrogen is the cause for the signals, in which case choosing the correct solvents is a non-issue, but capturing signals from the solute may be. Or it was figured out that using certain solvents gave ‘cleaner’ spectra, in which case it might also be interesting to know how the solvents were chosen.
I realise that there may be a good deal of speculation, retrofitting, inflationary use of the term serendipitously, anecdotical evidence, storytelling and bad memories involved, but is it known how the discoverers of NMR spectroscopy realised that a ‘signal’ is a hydrogen atom in the molecule observed?