How are these two molecules identical?
Sameer. The key to understanding this is that the two asymmetric carbons are indistinguishable because of the symmetry in the molecule. When you do your inversions on them, the molecule looks different, and should be an enantiomer, but after you do the 180 deg rotation, you have essentially switched them over (now the molecule looks the same as the first molecule, but only because you can’t differentiate the two asymmetric carbons). 

Imagine that you switched one of the terminal methyl groups with an ethyl, then try the same trick. Because you have broken the symmetry of the molecule, the 180 deg rotation doesn’t get you back to the start. Hence, this meso isomerism relies on the mirror plane of symmetry (or a point of inversion) existing in addition to multiple asymmetric centers.