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The ColorFunction of a SphericalPlot3D has six arguments, the first three being the $x$, $y$ , $z$ coodinates in $\mathbb{R}^3$. The fourth and fifth argument are the actual parameterization parameters of the surface and the last argument is the distance from the origin (the radius).

# (Slot) and & (Function) together allow to define anonymous function. #4 and #5 refer to the fourth and fifth argument. Here is (essentially) equivalent rewrite with Function in long form:

SphericalPlot3D[1, {θ, 0, π}, {ϕ, 0, 2 π}, 
 ColorFunction -> 
  Function[
   {x, y, z, u, v}, 
   ColorData["Rainbow"][Re[SphericalHarmonicY[5, 2, u, v]]]
   ], 
 ColorFunctionScaling -> False
 ]

Here is the example from the documentation:

GraphicsGrid[
 Partition[#, 3] &@
  Table[SphericalPlot3D[
    1 + Sin[5 ϕ]/10, {θ, 0, Pi}, {ϕ, 0, 2 Pi},
    PlotPoints -> 100,
    ColorFunction -> 
     Function[{x, y, z, θ, ϕ, r}, Evaluate[f]], 
    PlotLabel -> f, 
    Axes -> None], {f, {Hue[x], Hue[y], Hue[z], Hue[θ], 
     Hue[ϕ], Hue[r]}}],

 ImageSize -> Full
 ]

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