The goal of these omens is to act as a reminder. By eating all of these foods that have positive connotations, a person realizes that now is the time he needs to be asking for these good things, because now is the time he is being judged. As soon as the person realizes that now is the time that he is being judged, he will realize that omens alone will not be enough for his salvation, and that repentance is needed. Therefore, eating these omens, which are a reminder that now is the time for repentance, is extremely appropriate for Rosh HaShana.
Another reason given for why we eat these "omens" has to do with the "spirit" of the holiday of Rosh HaShana. If one looks over the prayers on Rosh HaShana, one will find that the basic theme is one of proclaiming the kingship and greatness of Hashem. Although Rosh HaShana is the day on which we are being judged, we do not make requests for sustenance, health, long life, etc.. We instead demonstrate how we have accepted Hashem as our king, and that we will listen to Him and follow His dictates.
Yet, asking Hashem for one's needs really is not so far out of the realm of proclaiming the kingship of Hashem. By asking Hashem for our needs, we vividly illustrate the fact that Hashem is our superior, and that we depend on Him for everything. In reality, if we asked Hashem for our needs, it would demonstrate how we are the subjects of Hashem, and that we have accepted Him as our king.
In order to reconcile these two realities, we have the omens. By eating these omens (and with some, saying the accompanying liturgy), we are covertly asking Hashem for our needs. We do not want to do such blatantly, as that is not in the strict spirit of the day. However, as it does demonstrate that we have accepted Hashem as our king, and today is the day we are being judged, we "ask" Hashem that we be remembered for a good year in a fashion that is not outwardly a request.
The Omens Themselves
The foods mentioned by the Gemora all are omens because the food itself symbolizes a good or because their names connote or sound similar to words that indicate some sort of good thing. These words are used in the liturgy which is said upon eating the food, known by its first two words "Yehi Ratzon..." "May it be your will...."
(For a listing of the appropriate Yehi Ratzon's transliterated and translated in to English, see Supplement to 38: Yehi Ratzon - Text and Instructions (I:38a).)
The first food that the Gemora mentions is gourds, or "K'ra." The word "K'ra" sounds both like the word for "read/proclaim" and the word for "tear." When we eat the gourd, there are two possible "Yehi Ratzons" that can be said. The first goes "Yehi Ratzon milfanecha... _sheyikaru'u lifanecha zechuyo'seinu" "May it be your will Hashem that our merits _ be read/proclaimed_ before you." The other is"...sheyikora g'zar de'nainu." "...that the decree of our sentence should be torn up." There are varying customs as to this Yehi Ratzon, and a prevalent custom is to recite both endings together in one Yehi Ratzon.
The second food mentioned is "Rubia," or fenugreek. The word "Rubia" sounds like the word "yirbu," the word for "increase." We therefore say a Yehi Ratzon that contains the request "may...our merits increase."
The word for the third food, "Karsi," leeks or cabbage, sounds like the word "kares," "to cut off/destroy." We therefore say a Yehi Ratzon that asks "may... our enemies be destroyed."
The word for the fourth food, "Silka" or beets, sounds like the "siluk," meaning "removal." We therefore say a Yehi Ratzon that requests "may our adversaries be removed."
The word for the fifth and final food "Tamri" or dates, sounds like the word "sheyitamu," "that they be consumed." Hence, we sat a Yehi Ratzon that implores "may... our enemies be consumed."
Another food that we eat is honey, because of its sweet taste. We dip Chalah (holiday loaves of bread) and apples in honey. Upon eating the apple and honey, we say a Yehi Ratzon that beseeches from Hashem that "You (should) renew us for a good and sweet year."
Why do we eat apples? The Maharil explains that we find in Bereshis 27 that Yaakov disguised himself as Esav in order to "trick" Yitzchok, his father, into giving him the blessings of the first-born, which Esav had previously sold to Yaakov. When Yaakov came close to Yitzchok, who was blind, Yitzchok noticed a sweet smell emanating from Yaakov. Yitzchok commented (Bereshis 27:27) "See, the smell of my son is like the smell of a field which Hashem has blessed." He then continued to bless him with blessings of wealth and power. According to many commentators, the "field which Hashem has blessed" refers to an apple field, and the smell of that apple field is also the smell of the Garden of Eden. The Vilna Gaon continues to explain that this incident with the blessings occurred on Rosh HaShana. It is therefore fitting to eat an apple, a fruit whose smell is associated with the blessings of Yaakov (which were given on this day) as well as with the Garden of Eden, on the day on which we ourselves want blessings.