Why can Sephardic Jews eat rice on Passover? Is this specified in the Torah somewhere?
This question ought to be, "Why can Ashkenazi Jews not eat rice on Passover?"
There are 5 grains that make Ḥametz when mixed with water and allowed to rise: barley, rye, oats, wheat, and spelt. Any others that have been added by communal custom are just that - additions by virtue of communal custom.
Kitniyot are not specified in the Torah at all. They are chumrot — binding customs of the community.
The Sefardic kitniyot custom permits rice provided it has been carefully checked grain-by-grain before Passover begins to ensure that no chametz grains are in it. (How carefully depends on the tradition; I have seen "3 times" and "7 times" specified by different traditions.) Ashkenazic custom is much more restrictive when it comes to kitniyot.
Chametz is a biblical term which refers to grains which ferment when water is added. This fermentation causes the dough mixture to puff up. The talmud (Pesachimn 35a) discusses the fact that rice and millet also puff up when water is added.
The structure of the discussion:
The Mishna lists the grains that one may use for the mitva of eating matza on the first night of pesach: Wheat, barley, spelt, rye, and oats.
The gemara brings tannaic sources that the list is derived from a biblical juxtaposition of chametz and matza- only grains which "ferment" (become chametz) may be used for matza, but not those which "rot".
The gemara continues to underscore that the above mishna rejects the opinion of the tanna Rabbi Yochanan ben Nuri, who classifies rice and millet as grains which ferment.
From the above gemara, there seems to be a tannaic dispute as to what state of fermentation is called "chametz". Rabbi Yochanan ben Nuri has a broader definition of the term chametz, but his minority opinion is rejected from halacha in favor of a more specific type of fermentation which would categorize rice as non-chametz.
Communities which refrain from eating rice on Pesach are limited to those communities who have accepted to do so.