Why can’t women lein?
Nedarim 37b states quite clearly that girls should be taught Torah with the trop. (This makes sense; the trop is another level of nuance in understanding the meaning of the text.)
(Near the top of the column.)
Megillah 23a states that women are eligible to be called up to the Torah, but that Chazal said it shouldn’t be done because of “kavod hatzibur.”
(About 2/3 down the page)
What that means, and what the nature of the restriction is, is the debate. Conservative Judaism doesn’t consider it a halacha or takanah. Orthodoxy assigns more importance to it than many other Talmudic statements.
Partnership minyanim are Orthodox minyanim that permit women to lein on the basis that a tzibur can congregate with the specific intention of being mochel on its kavod. However, approximately no Orthodox rabbis actually support this notion, although their arguments are rarely halachic in nature.