Kashrut is a system of separations rather than a list of banned foods, and the separations do most of the work.
Permitted land animals must both chew the cud and have cloven hooves. Permitted sea creatures must have fins and scales, which excludes every shellfish. Birds are permitted by inherited tradition rather than by a stated rule, so the list is received rather than derived, and novel species raise real questions. Blood is prohibited absolutely, which requires meat to be drained and salted before use.
The separation of meat from dairy is a rabbinic elaboration of a single verse forbidding the boiling of a kid in its mother's milk, and it is the rule that shapes a kitchen. A kosher household maintains two sets of crockery, cutlery and pots, frequently two sinks and two dishwashers, and observes a waiting period of between one and six hours after meat before dairy may be eaten, the interval varying by community and inherited from parents. Anything neither meat nor dairy is neutral and may accompany either, which is why fish and eggs occupy a category of their own.
Passover adds a second system on top of the first. Leaven is not merely avoided but must be absent from the property, so the kitchen is cleaned, remaining leaven is formally sold to a non-Jew for the duration, and a further set of dishes is used.
The reasons given have varied enormously. Hygiene explanations are popular, modern, and unsupported by the texts, which mostly give holiness and separation as the reason and decline to explain further. Maimonides offered health arguments. Other authorities insisted the laws are decrees whose whole point is obedience precisely because no reason is supplied.