A rationalist explanation of simana milsa by R. Yehuda Prero (link):
The Gemora in the tractate of Kerisus (6a) states "Abaye said 'Now that you have said that an omen is significant, at the beginning of each year, each person should accustom himself to eat gourds, fenugreek, leeks, beets and dates...'." Because of this Gemora, it is a custom to eat these listed foods, as well as other foods, which represent good things. (We will soon explore how exactly these foods are representative of good things.) The issue that must first be addressed is why do we "indulge" in omens at the beginning of the year, on Rosh HaShana? As we will soon see, there are many, many different omens and customs. Why do we eat these foods on this occasion?
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... 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.