I'm not sure where the "not interest" part of your quoted statement comes from; that's not the way this advice is normally given. Perhaps you worded the statement that way because you think there are only two possibilities: payments either are applied to the principal or to the interest? If so, there's a misunderstanding here.
Here are two other possibilities: (1) the mortgage holder puts the extra payments in escrow/holding, to be used in the future should you fail to make a payment; (2) the mortgage holder records the payment as being made in advance, without giving you any benefit (such as reduced interest). (Another way of thinking of this is that the mortgage holder moves up the due date, essentially giving you credit for having made the final payment. Unfortunately, this isn't just theoretical.)
In short, "applied to principal" is a shorthand way of saying that the mortgage holder treats the extra payment in a particular way (which benefits you the most), and not in a way (see (1) and (2), above) that benefits the mortgage holder the most.