For each of the plots of the Johanson antenna, when measuring the radiation pattern, they measured it with a dual polarized reference antenna, showing the output power in V and H respectively. The XZ plane shows isotropic radiation for vertical polarization, and some deep nulls in the horizontal polarization; this tells me that if you're designing this antenna into a product, position the antenna so that the XZ plane intersects the other station, assuming the other station is transmitting vertically polarized. This is often the case for WiFi and other 2.45 GHz applications. If your radio has a 'diversity' receiver, add a second antenna with the XY plane intersecting the other station, since its 'vertical' pattern is higher overall than the pattern in the YZ plane.
Also compare the XY-H and YZ-V plots. From the geometry, they should both be pointing at the 'top' of the antenna (marked with the dark square) at zero degrees, and in both cases, the gain is about -25 dBi. The change in polarization results from the 'twist' of the antenna-under-test with respect to the receiving (reference) antenna. Similarly, XY-V and YZ-H have the same value of about -10 dBi at 0 degrees. You can do similar comparisons between the other plots as well to help understand the pattern in 3D, and the relationship between the patterns in two polarizations.
Hope this helped. Also, I agree with hagster, the plots in 3D just look cool; to me the 2D cuts convey a lot more information.