Incorrect: The acquaintances who we like most are those that flatter us best.
Correct: The acquaintances whom we like most are those who flatter us best.
Why could not we say:
-> The acquaintances that we like most are those that flatter us best.
I do not find the MGMAT's explanation very clear on this one.
The word "acquaintances" is the key. You see it is a plural so "that" is not appropriate, may be "those" can be used. But then "those" is already used in the later part of the sentence and for me "those" doesn't fit in well in that place like "whom". As far as I know "whom" is a common reference for both singulars and plurals. Not really sure if this is right, just my thoughts.