Really functions here as an emphasising device, rather than as conveying any precise information. In 'I am really not into it', really qualifies not and puts emphasis on it; the sentence is thus roughly equivalent to 'I am not into it', where not is uttered or printed with special emphasis. On the other hand, 'I am not really into it' is a negation of 'I am really into it'. One's purpose in negating it may be no more than to remove the emphasis conveyed by really; the removal of that emphasis is compatible with one's being somewhat into it.

Several other answers have explained the difference in meaning between the two sentences, but it should also be made clear how the syntactical differences between them generate that difference in meaning. Really functions here as an emphasising device, rather than as conveying any precise information. In 'I am really not into it', really qualifies not and puts emphasis on it; the sentence is thus roughly equivalent to 'I am not into it', where not is uttered or printed with special emphasis. On the other hand, 'I am not really into it' is a negation of 'I am really into it'. One's purpose in negating it may be no more than to negate the emphasis that would be conveyed conveyed by really; the negation of that emphasis is compatible with one's being somewhat into it.

The first thing to note is that really functions here as an emphasising device, rather than as conveying any precise information.

  In 'I am really not into it', really qualifies not and puts emphasis on it; the sentence is thus roughly equivalent to 'I am not into it', where not is uttered or printed with special emphasis.

  'I am not really into it' is a negation of 'I am really into it'. One's purpose in negating it may be no more than to remove the emphasis conveyed by really; the sentence is thus compatible with one's being somewhat into it.

Really functions here as an emphasising device, rather than as conveying any precise information. In 'I am really not into it', really qualifies not and puts emphasis on it; the sentence is thus roughly equivalent to 'I am not into it', where not is uttered or printed with special emphasis. On the other hand, 'I am not really into it' is a negation of 'I am really into it'. One's purpose in negating it may be no more than to remove the emphasis conveyed by really; the removal of that emphasis is compatible with one's being somewhat into it.