The ultimate goal of Yom Kippur as described in Parshat Acharai Mot is to achieve כפרה, atonement for one's sins. We achieve this goal through עינוי נפש, affliction of the soul. The Chachamim define affliction as abstinence from five activities, including eating and drinking. The connection between a physical deprivation and deprivation of the Nefesh, the soul, is somewhat puzzling since only a גוף, a physical body, requires physical sustenance. Furthermore, how does physical depravation affect כפרה? There appears to be at least one missing step.
The root ענה in addition to meaning depravation, can also mean to speak or to raise one's voice, as it states in Parshat Ki Tavo in conjunction with מקרא ביכורים, וענית ואמרת לפני ה' אלוקיך. The אור החיים suggests an additional meaning for ועיניתם: make yourselves lowly (or poor) before Hashem. In this way we recognize that everything is from Hashem.
Perhaps we can combine these two interpretations. ועיניתם את נפשותיכם will then mean, "And you shall answer your inner soul." You will speak to your soul and realize how deprived it has been. Subsequently, you will engage in Teshuva and verbally declare Vidui, as the Rambam requires (Hilchot Teshuva 3:3), "When he does Teshuva he must confess with his mouth."