This is a great question, because it highlights one of those cases (which I'm sure must be very annoying to English learners) where a word can mean two different, opposite things.
The meanings of the word spare which are intended here are:
4 : healthily lean
5 : not abundant or plentiful
In other words, the author is saying they are putting up with a frugal or minimal (rather than plentiful) diet. You are right that this is, in some sense, the opposite of the more common meanings:
1 : not being used
2 : being over and above what is needed