How did the tradition start?
Why is it still around today?
What is the basis for this Halacha (הלכה)?
There are sources beginning in the 1200s (the Mordechai and others) that quote the practice to avoid various legumes and semigrains; either because of concern that they make contain some wheat (or other chametz-causing grain) mixed in; or because if you grind them into flour, people may think you're using wheat flour (or barley spelt etc.) and making chametz. See this question for more details. Rabbi Joseph Soloveichik famously observed that a Rabbi Yohanan ben Nuri had prohibited rice in the Talmud; thus the custom has at least some minority Talmudic opinion on which to hang its hook (even though for entirely different reasons).
The custom against kitniyot is codified as Ashkenazic practice by Rabbi Moshe Issreles in his comments on the Shulchan Aruch (c. 1550).
As it has become established custom, it is now binding on Ashkenazic Jews, barring rare circumstances (someone could ask their rabbi for a waiver if they have severe dietary restrictions; in times of severe scarceness, some communities' rabbinic leadership would allow some forms of kitniyot that year). Rabbi Danczig, writing in the Chayei Adam right about 1800, explains that it was not instituted top-down by rabbinic leadership (in which case it would warrant the Mosaic prohibition "do not deviate from what your judges state, left or right"), but something that happened bottom-up, in a softer, more organic fashion, and has now been fixed. He applies the opening verse of Proverbs, "don't reject the teachings of your mother." Around the year 1900, Rabbi Yechiel Michel Epstein of Novardok (in Aruch HaShulchan) wrote that "those Ashkenazic Jews who fail to keep kitniyot today, it is clear that they lack all fear of Heaven."
We view standardized custom as something very strong and not something that should be tinkered with. We don't think we're so smart that 800 years of Jewish history can just be thrown out the door; some caution and humility are in order.
There are, however, a few loopholes:
As per Rav Moshe Feinstein's approach, that the prohibition to eat kitniyos is a hanhagah shehinhugu chachamim, a custom that the Rabbis guided their flocks to keep. It does not have the status of a takannah, a law instituted by court. All the more so is this true according to the Chayei Adam's approach mentioned by Shalom.
Since it was based on a good reason it has the status of a neder, a vow to do good by not using these products on Passover. For such a vow that was taken on by one's ancestors for a d'var mitzvah, (to keep us further away from the possibility of eating hametz) the law is that even if it is more difficult for us to keep it, we must because of the verse "Shemah beni musar avichah ve'al titosh toras imecha" "... do not leave the guidance of your mother" i.e. One must not deviate from the good guidance and customs that his ancestors accepted on themselves. Being that an Ashkenazi must keep this custom however difficult it may be (except if endangering to life) it would appear unmovable even by a Sanhedrin. The only way to remove a vow is through finding an opening for regret, which is an impossible exercise if you did not make the vow to begin with.
Those Ashkenazi gedolim who did not like the custom lived in times when it was either not unanimously accepted yet in the Ashkenazic world and they wished to keep it that way (some Rishonim), or it was recently accepted by a previous generation and the gadol was expressing frustration that he could not do away with it (Rav Yakov Emden).