The word comes from the expression
"non me ne frega niente", unpolite version of
"non me ne importa nulla" (I don't care at all);
menefreghismo is perfect Italian (not unpolite) and the meaning is:
"the attitude of not caring about something (that maybe you should care of)".
For instance: "menefreghismo per la politica",
"menefreghismo per lo stato della Scuola", etc.;
It is different from 'disinteresse', as a subtle blame is implied in this word, like
"questo medico mostra menefreghismo per le sorti dell'ospedale";
compare with:
"mio fratello ha un totale disinteresse per il calcio"