In reading Galois theory, I have found it especially useful to keep in mind the historical questions that motivated its construction (the Wikipedia article gives a nice summary of the kind of context that I found helpful).
What are similar motivating questions that I can keep in mind when reading real analysis, in particular while reading Baby Rudin? (Baby Rudin covers a lot of topics. I'd find context most useful for the elementary topology that's used (set properties such as open/closed, compactness, Cauchy sequences).)
Applications are also useful (e.g. cryptography was good to keep in mind for group theory and Galois theory) but I am looking more for motivating mathematical context. Not "why do people care?" but "why did people care?"
[In particular, this context is helpful to avoid searching for structure that doesn't exist. For instance, in Galois theory we have definitions involving general automorphism groups, but in practice maybe we only care about a common subset of them that we can reason about. I'd be interested to see similar answers for other areas of mathematics as well.]