Please help me with using ListDensityPlot

Edit

data;
newdata = 
  Function[{χ, ϕ, 
     z}, {(χ Degree)*Cos[ϕ Degree], (χ Degree)*
      Sin[ϕ Degree], z}] @@@ data;
ListDensityPlot[newdata, PlotLegends -> Automatic, 
 FrameLabel -> {"χ (°)", "Φ (°)"}, 
 BaseStyle -> {FontSize -> 18, FontWeight -> Plain, 
   FontFamily -> "Helvetica"}, InterpolationOrder -> Automatic, 
 BoundaryStyle -> Directive[Thick, Black], 
 RegionFunction -> Function[{x, y}, x^2 + y^2 <= .2^2], 
 AspectRatio -> Automatic]

Original Maybe something like this.

data = Flatten[
   Table[{χ, ϕ, Sin[χ*ϕ]}, {χ, 0., 4, 
     0.1}, {ϕ, 0., 2 Pi, 0.1}], 1];
newdata = 
  Apply[Function[{χ, ϕ, 
     z}, {χ Cos[ϕ], χ Sin[ϕ], z}], data, 1];
ListDensityPlot[newdata, PlotLegends -> Automatic, 
 FrameLabel -> {"χ (°)", "Φ (°)"}, 
 BaseStyle -> {FontSize -> 18, FontWeight -> Plain, 
   FontFamily -> "Helvetica"}, InterpolationOrder -> Automatic, 
 BoundaryStyle -> Directive[Thick, Black]]

Thank you for posting your complete data. You can transform your data into cartesian coordinates easily. Assuming that your dataset is data, then:

transformed = 
  Function[{rho, theta, intensity}, {rho Cos[theta], rho Sin[theta], intensity}] @@@ data;

This can be digested by ListDensityPlot directly. Below I have added a sigmoidal shaper to the ColorFunction, acting directly on your intensity values. You should play around with the intensityThreshold until you find a value that you think best highlights the features of your data set. You might want to use Manipulate for this as well.

With[{intensityThreshold = 12},
  ListDensityPlot[
    transformed,
    ColorFunction -> (Tanh[# - intensityThreshold] &),
    ColorFunctionScaling -> False,
    ImageSize -> Medium,
    Epilog -> {Thickness[0.01], Darker@Green, 
               Circle[{0, 0}, Max[transformed[[All, 2]]]]},
    Axes -> False, Frame -> False
  ]
]

The data represents intensity measurements on the surface of a sphere. The pole-plot is a view of the surface of the sphere looking at the origin from the positive z-axis. For ListDensityPlot, we need to project points on the surface of the sphere (blue point) onto the x-y plane (red point).

FromSphericalCoordinates gives the x,y,z coordinates. Ignore the z-axis because ListDensityPlot needs only the x-y coordinates. We find this expression for the x and y coordinates:

FromSphericalCoordinates[{1, [CapitalChi], [CapitalPhi]}][[1 ;; 2]]

{Cos[Φ] Sin[?], Sin[Φ] Sin[?]}

The first two columns of the data are degree measurements. Convert these two columns because Sin and Cos require radian values. Find the spherical coordinates projected onto the x-y plane. Borrowing from MarcoB's answer, plot the intensity measurements with ListDensityPlot, with radius circles at 15° intervals (15, 30, 45, and 60), and radial grid lines.

data[[All, 1 ;; 2]] *= Degree;
(*data projected onto the x-y plane*)
xyData = Function[{[CapitalChi], [CapitalPhi], intensity},
  {Cos[[CapitalPhi]] Sin[[CapitalChi]], 
    Sin[[CapitalPhi]] Sin[[CapitalChi]], intensity}] @@@ data;

intensityThreshold = 12;
ListDensityPlot[xyData,
 ColorFunction -> (Tanh[# - intensityThreshold] &), 
 ColorFunctionScaling -> False,
 Axes -> False, Frame -> False,
 Epilog -> {Thickness[0.01], Darker@Green, Circle[{0, 0}, 1],
   Thickness[0.005], Dashed,
   Table[Line[{p, -p}], {p,Table[{Cos[a], Sin[a]}, {a, 
     Subdivide[0, 150, 5] Degree(*30° increments*)}]}],
   Table[Circle[{0, 0}, Sin[d Degree]], {d, 15, 60, 15}]}]