A private exhibition is one to which specific individuals are invited, as opposed to a public performance which is open to anyone, or perhaps to anyone who buys a ticket. A private exhibition will not usually charge admission, either, although some public performances are free of charge, so that cannot be the criterion.
An ordinary commercial theater showing is a public performance.
In this case, however, the further restriction "in homes" is added, so only private exhibitions in a private home are authorized. Also, the limitation "non-commercial" is added, so any gathering that charges admission will not be in accord with the license.
In short, such recordings may be played for a gathering of family and friends in a private home, with no charge for admission, but not for some event open to the public, nor for any event in a public space, as opposed to a home.