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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
]