The Jewish nation was founded both individually and collectively in a miraculous environment. Yitzchak was born of parents regarding whom the Torah testifies, "now Avraham and Sarah were old, well on in years, the manner of women had ceased to be with Sarah" (Braishis 18:11). The enslaved nation, even prior to the exodus, witnessed the wonders of Hashem in Egypt in conjunction with the ten plagues. They were escorted out of Egypt with the divine cloud by day, and by night with a pillar of fire. They walked through water on dry land and witnessed the Egyptians drown. At Marah, the bitter waters turned sweet by throwing a bitter tree therein, and seventy date palms granted them as they entered the desert.
The daily miraculous ration of mon that nourished them rotted if kept overnight for the morrow, except Shabbos, when they were commanded to take a double portion which retained its freshness. The menorah was fashioned miraculously out of fire, as was the briach hatichon - the one middle bar inside the planks of the mishkan kept all the planks secure. For forty years they were surrounded by constant miracles. The great importance and significance of these miracles, teaches the Ramban in parshas Bo, is to impress upon the Jewish nation the close personal relationship that Hashem has with Bnei Yisroel. His hashgacha pratis - Divine intervention and concern for their welfare - is the cornerstone upon which the Jewish people is founded.
Sadly, but realistically, the nature of man is to forget. Not only do later generations forget the impact of miracles, but even the one to whom the miracle occurred is wont to forget. Case in point: when King Shaul hesitates to allow Dovid to fight Golyas, Dovid defends himself by sharing his prowess in defending the sheep from the attack of both a lion and a bear. Dovid said, "v'nasah she meihaeider - the lion or bear would carry off a sheep" (Shmuel I 17:34). The word is written "seh"- a sheep - but is read "zeh - this". The Vilna Gaon (in Kol Eliyahu) explains based upon the medrash, Dovid slaughtered the sheep after he saved it, and from its skin he made a vest which he wore constantly, enabling him to remember the miracle always. If not for the vest, Dovid would have forgotten. It is the "zeh" that he showed Shaul. Rav Yerachameal Kron shlit"a in his sefer V'Talmudo B'Yado notes that while Moshe successfully prays for the recovery of his sister Miriam, he does not pray on his own behalf to have his speech impediment cured, as Moshe wanted to always remember the miracle of the angel directing his infant hand to the hot coals rather than the alluring gold (Yalkut Shimoni, Shemos #166). Had his speech defect been removed, he too may have forgotten the miracle.
If the direct beneficiaries of the miracle is prone to forget it rachmana litzlan, all the more so their descendants and future generations. To prevent this from happening, to keep the memory of the miracles fresh in the minds of all generations, explains the Ramban (ibid), many mitzvos have as their primary purpose "zecher l'yitzias mitarayim" - to remember the exodus from Egypt and its accompanying miracles and wonders. To offset the heresy that Hashem is removed from the activities of mankind, Hashem suspends nature, and has us incorporate His miraculous involvement as an integral part of many mitzvos. Thus, our donning tefilin, mezuzos on our doors, the many prohibitions of chometz coupled with the mitzvah of eating matzah, all keep the memory of the miracle alive and vibrant. Our annual immigration to the sukkah is explained by the Torah as "l'maan yadeuh dorosaychem" (Vayikra 23:43) - so that your generations will know that millions of people were fed, clothed, and sustained miraculously for forty years in a desert environment.
The Torah (Devarim 6:16) warns "lo senasu - you shall not test Hashem". The Ramban understands this to mean you are not to say that if Hashem is indeed in our midst, let Him prove Himself by performing miracles on our behalf. This is wrong on several levels, including a lack of trust and faith in the past performances and the inherent deficient manner of serving Hashem - based upon a reward. Moreover, while miracles are performed, it is not the desire of Hashem to perform open miracles regularly, and it is the way of Hashem most often to minimize the extent of the miracle. Thus, technically, notes the Ramban (Braishis 6:19), even many arks the size of Noach's ark would not suffice to house all the animals and their provisions for a year's time. Yet, by having Noach build a relatively large ark, the miraculous was couched within the natural order of things.
The Steipler zt"l in his sefer Chayey Olam suggests an interesting reason why miracles were the order of the day at the foundation of our people. They needed to learn the very important lesson of our connection to Hashem. However, the presence of open miracles diminishes the free will of man. In addition, having witnessed open miracles the level of expectation and accountability for man is raised, and even smaller infractions are treated more seriously. Given the nature of man to forget the miracles, it can only serve to create obstacles on man's behalf.
In the bracha of modim, we recite thrice daily, "al nisecha shebechol yom imanu", we thank Hashem for Your miracles that are with us every day. It includes the natural phenomena around us most often taken for granted, such as the burning of oil. As R' Chanina said (Gemara Taanis), "the one who endowed oil with the ability to burn can cause vinegar to burn". It also includes the survival of the state of Israel, surrounded by multitudes of unfriendly Arabs. It includes His hasgacha pratis - direct involvement in our personal and communal lives as the many mitzvos of zecher l'yitzias Mitzrayim proclaim.
While belief in miracles is not one of the 613 mitzvos, it is the theme, and at the heart of, many mitzvos.