Lattices like intervals or sets of values or zero/nonzero are typical and natural even without studying lots of theory.
I believe this paper is about how you can use the concept of a particular kind of lattice to give a rigorous mathematical semantics to possibly non terminating computations such as the untyped lambda calculus. This was not at all obvious before Scott. I don't feel qualified to comment much beyond that.
I suppose this is related to the simpler examples of lattices, but it's quite a feat.