Possible Products in the Addition To a Double Bond in a Ring
why should the plane in which both the hydrogen atoms are present even
  matter?


In this case it doesn't.  The author of the example is laying the groundwork for the future.

The area above and below the double bond is often referred to as a "face". In your molecule, the two faces are not equivalent. If you build two models of your compound and in both color the top of the double bond red and the bottom blue, you'll find there is no way to superimpose one model onto the other where the red face is superimposed on the blue face.  These two faces are therefore not equivalent, they are said to be enantiotopic (mirror images) in this molecule.

In this case, addition of (an achiral reagent such as) hydrogen to the two faces produces the same molecule.  In more complex molecules, or when using chiral reagents this will not be the case.  

Consider the case where there is a methyl group attached to the methylene group adjacent to the $\ce{C-Br}$ carbon in your molecule.  Now the faces are different (diastereotopic) and addition of hydrogen to the top and bottom faces will produce two different products (methyl and bromine on the same side or opposite sides of the molecule).

The concept of faces is important in chemistry.  Always consider if the faces are equivalent, enantiotopic or diastereotopic. Depending on what you find, the products from attack at the two faces may be the same, enantiomers or diastereomers.  The concept of faces is introduced in your problem.