Scott-continuity emerged when Dana Scott build the first model of untyped λ-calculus, while trying to prove that no such model can exist (since any such model $D$ needs to be, simplifying a bit, isomorphic to the function space $D \rightarrow D$ which is not possible set-theoretically, but turns out to be possible when you restrict your attention to computable functions).
Scott-continuity can be understood as a mathematically well-behaved approximation to computability.
 is a gentle introduction to the general area of order theory that Scott continuity emerged out of, and  is a reference article.  has a bit on domain-theory and Scott-continuity and might be the easiest introduction for computer scientists.
B. A. Davey, H. A. Priestley, Introduction to Lattices and Order.
S. Abramsky, A. Jung, Domain theory, https://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf
G. Winskel, The Formal Semantics of Programming Languages: An Introduction.