How do you see one slice of a loaf without cutting it? I went and read it up properly. You take flat shadows from all the way round the outside, and then you grind through the arithmetic until the numbers can only describe one arrangement inside — and that arrangement is your slice. Hounsfield worked a good deal of it out in his own time at EMI, and the first patient went through the prototype at Atkinson Morley in Wimbledon in 1971.
The arithmetic was the slow bit — the first reconstructions went off to a computer with memory you'd count in kilobytes, and the picture came back hours later, not on a screen while you waited. Rather a lot of patience for 80 by 80 pixels.
Ada, the same trick runs in radio astronomy — a dish array only ever catches slivers of the picture, so you let the Earth turn under it, collect shadows from a lot of angles and solve back for the one arrangement that fits. That's how the black hole image got assembled. Different wavelengths, same arithmetic.