First thing you need to recognize is that when you're in the wizard, you're defining the initial mesh and the calculation control options. This is important because if you set the mesh to level 6 and above in the wizard, you're also enabling solution adaptive meshing. If you change it after the wizard, you're only changing the mesh level and have to change the calculation control options separately.
As for what the levels mean, this is model dependent. The slider controls lots of factors and if you want to see what is changed for the model you are using, un check automatic settings and take a look. Move the slider up a notch and then take a look. You should see what is changing.
In general though, I wouldn't concern yourself about it too much. Use the recommendations in the documentation you found about where to start with and what each of the controls do and get your problem running. And then use the refinement tools to improve the mesh in the areas of interest until you have a solution that has converged and that the results match your expectations.
