They literally sliced it. With a razor blade or scissors, in a splicing tool. Then taped it back together.
To do things like a dissolve (where scene A fades out and scene B fades in), you would project scene A with decreasing light and scene B with increasing light onto a new strip of film.
Audio was done separately on its own big piece of tape using similar processes, then the finished audio and video tapes would be combined in the final stages.
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