The Talmud in Kritot 6a tells us that certain foods are customarily eaten on Rosh Hashanah. Abaye said, “Now that you have determined that omens are significant, at the beginning of every year a person should accustom himself to eat gourds, fenugreek, leeks, beets, and dates …”
The custom developed not just to eat the foods, but also to place our hands over them and to recite a prayer for the upcoming year over these foods. The prayers would be clever play on words, so that the food eaten actually symbolized what we want from the new year. We are kind of eating our prayers!
The Avudraham says: "On the gourd (krah) they would say: “Our bad decrees should be torn up” (yikrah); on the fenugreek (rubya) they would say: “Our merits should be numerous” (yirbu); on the leek (karti) they would say: “Our enemies should be cut off” (yikratu); on dates (tamarim) they would say: “Our sins should be removed” (yitamu). They would then add a pomegranate and say: “Our merits should increase like (the seeds of) a pomegranate.” Some would preface, “May it be Your will, Lord our God and God of our forefathers …”
Rabbi Menachem Meiri in Chibur Hateshuvah says that we do this in order to focus ourselves on the importance of the day even during the meal. In order not to allow our practices to seem like sorcery, the custom began to actually say words of prayer over the simanim. The sages were concerned that we might just spend the entire day eating and not focusing on the spiritual service of the day, so prayers were added to remind us that judgement is happening all the time.
Even when we ask for our enemies to be "cut off" "yikartu" we are, says the Meiri, really praying that foreign ideas that can draw us away from our service to G-d be cut off from our lives.
Some people I know even create their own simanim. They take raisins, and crush them, and serve with celery sticks for dipping into them. They are of course praying for a 'raise in salary'! Or some are careful to eat figs, so that they are asking G-d to help them 'fig-ure' out what to do with their lives this year! Of course all this comes to allow us to feel close to the greatness of the day.