The key is the connection between Galois groups and groups of deck transformations. The Galois group of the general quintic is S₅, while ℑ: ℂP¹ → ℂP¹ is a branched covering whose group of deck transformations is A₅.

For a modern treatment see Jerry Shurman's free book Geometry of the Quintic, and also this:

• Oliver Nash, On Klein's icosahedral solution of the quintic.