Frank Moore Cross writes about the subject in "Canaanite Myth and Hebrew Epic". It's not that a religion disappears, but rather that a particular god gets assimilated or replaced by another. For example, we have El Elyon who gets assimilated by Yahweh. Baal also gets assimilated (god of the storm) and replaced. Old elements of the cult become forbidden (no graven images), so the old gods either fall out of use or just become forbidden by law.

Another way a god disappears is when it's completely replaced by an equivalent deity or religious figure; an example is greek/roman gods being replaced by Catholic saints, like St. Christopher replacing Atlas, or indigenous deities being replaced by different advocations of virgin Mary, like La Virgen de san Juan de los Lagos, la Virgen de la Caridad del Cobre, la Virgen de Guadalupe, etc.

Other ways religions disappear is by military conquest, like in Mexico City: The Spaniards forbid worship of old gods, their temples get destroyed and new temples replace them. The Metropolitan Cathedral in Mexico City was built atop an Aztec sacred precinct near El Templo Mayor, after this one was buried.

Another way religions change is syncretism: Old gods get merged with the new, and new religions can arise. An example is the Santeria, where people use images of Catholic saints to represent pagan deities.

So, basically, the answer to how old religions disappear, is that they're replaced either partially or totally by new ones.