There is a hint from the text that the עינוי referred to involves an abstention from food.
The parsha both in Emor and in Pinchas lists all the holidays and says with regard to each "כל מלאכת עבדה לא תעשו," "all work of labor you shall not do." Except for Yom Kippur. There it says "וכל מלאכה לא תעשו," and all work you shall not do (Lev. 23:28; Num. 29:7). The only other holiday or time-period during which the Torah invariably forbids "all work" as opposed to just "work of labor" is Shabbos (E.g. Lev. 23:3 "כל מלאכה לא תעשו," all work you shall not do).*
What kind of work is "work" but not "work of labor"? I.e. what kind of work is not laborious enough to be prohibited on most holidays but is prohibited on Shabbos and Yom Kippur?
The answer is work related to food. On the one hand, the Torah expressly qualifies its prohibition of work on Pesach -- one of the regular holidays during which only "work of labor" is prohibited -- by saying "אך אשר יאכל לכל נפש הוא לבדו יעשה לכם," "only that which to every life is food, that alone shall be done for you" (Ex. 12:16). That is, work related to food preparation is excluded from the prohibition. On the other hand, with respect to Shabbos the Torah gives an express prohibition against the quintessential food-preparation work -- "לא תבערו אש בכל משבתיכם ביום השבת," "you shall not burn fire in all your dwelling-places in the day of Shabbos. (Ex. 35:3)" It seems what is going on is that the difference between "all work" and "all work of labor" is food-preparation-related labor, which is included in "all work" but not in "all work of labor."
It turns out then that work related to food preparation is prohibited on Yom Kippur. The question is why. Shabbos we have what to work with; we know it is a commemoration of Hashem's rest from the six days of creation (see Ex. 31:17), and making fire and food are creative acts. But why Yom Kippur?
Well, one thing we know about Yom Kippur is that there is an obligation to cause עינוי to the נפש; pain or subjugation to the physical life-force. Now perhaps this could be interpreted in more than one way, but abstention from food is certainly a valid possibility. It is a subjugation of one's natural physical drive. And we now have corroboration for this possible interpretation. Why else would the Torah forbid work related to food-preparation if not because this is in fact the correct interpretation of ועניתם?
*There are places where the Torah describes the prohibition on work on holidays as being a prohibition on "work" generally, but it seems logical to assume that the specific reference to "work of labor" in the detailed holiday lists of Emor and Pinchas clarify and qualify those general statements elsewhere. Yom Kippur and Shabbos, on the other hand, are never associated with a prohibition on mere "work of labor."
