Personally, I always thought the point was a fever makes it hard to keep anything down, so it’s better to just starve it. But I probably made that explanation up out of my own imagination.
Some years ago I had a friend who researched sayings like this, and what I heard from her went as follows:
The modern form of the saying is “starve a cold and feed a fever”.
The original version of this saying was “if you starve a cold, you will be feeding a fever”.
In other words, if you do not get proper nutrition when you have a cold, then you could get even more sick (will be feeding a fever).
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Did your friend happen to mention where she got the information about the original phrase?