Are a meso compound and its mirror image isomers? How are meso isomers even isomers? By definition, meso isomers are compounds that are superposable on their mirror images, but doesn't that mean that the mirror image is just the same compound?

Can someone explain the concept of meso isomers and diastereomers to me?
meso-compound
A term for the achiral member(s) of a set of diastereoisomers which also includes one or more chiral members.[1, 2]
isomer
One of several species (or molecular entities ) that have the same atomic composition (molecular formula) but different line formulae or different stereochemical formulae and hence different physical and/or chemical properties.[1, 3, 4]

Your notion is correct in so far as the meso-compound and its mirror image are not isomers of each other since they are actually the same compound.
For example, meso-tartaric acid (preferred IUPAC name: (2​R,3​S)-2,3-dihydroxybutanedioic acid) can be superimposed on its mirror image “(2​S,3​R)-2,3-dihydroxybutanedioic acid”; i.e. both structures actually describe the same compound. Thus, they are not isomers of each other. (And hence, “(2​S,3​R)-2,3-dihydroxybutanedioic acid” is not a correct IUPAC name.)
  
However, two further stereochemical configurations are possible, namely (2​R,3​R)-2,3-dihydroxybutanedioic acid and (2​S,3​S)-2,3-dihydroxybutanedioic acid.
  
These structures are non-superimposable mirror images of each other, i.e. they are enantiomers, and the above-mentioned meso-tartaric acid is a diastereomer of both of them.
Therefore, there are a total of three isomers of tartaric acid:

(2​R,3​R)-2,3-dihydroxybutanedioic acid,
(2​S,3​S)-2,3-dihydroxybutanedioic acid, and
the meso-compound (2​R,3​S)-2,3-dihydroxybutanedioic acid.

References
[1] Basic terminology of stereochemistry (IUPAC Recommendations 1996). Pure Appl. Chem. 1996, 68, 2193
[2] meso-compound. In IUPAC Compendium of Chemical Terminology, (Gold Book)
[3] Glossary of terms used in physical organic chemistry (IUPAC Recommendations 1994). Pure Appl. Chem. 1994, 66, 1077
[4] isomer. In IUPAC Compendium of Chemical Terminology (Gold Book)