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The ColorFunction of a SphericalPlot3D has five arguments, the first three being the $x$, $y$ , $z$ coodinates in $\mathbb{R}^3$. The last two are the actual parameterization parameters of the surface.

# (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
 ]

Compare this also to the following:

g = GraphicsRow[   
 Table[
  SphericalPlot3D[1, {θ, 0, π}, {ϕ, 0, 2 π},
  PlotPoints -> 100,
  ColorFunction -> f
  ],    
  {f,
    {
     Function[{x, y, z, u, v}, ColorData["Rainbow"][(1 + Sin[5 Pi x])/2]],
     Function[{x, y, z, u, v}, ColorData["Rainbow"][(1 + Sin[5 Pi y])/2]],
     Function[{x, y, z, u, v}, ColorData["Rainbow"][(1 + Sin[5 Pi z])/2]],
     Function[{x, y, z, u, v}, ColorData["Rainbow"][(1 + Sin[5 Pi u])/2]],
     Function[{x, y, z, u, v}, ColorData["Rainbow"][(1 + Sin[5 Pi v])/2]]
     }
    }    
    ],   
  ImageSize -> Full   
  ]

enter image description here