I don't mean to sound silly, but what does a sea strainer do?

It cleans (strains) the debris out of the water going to the raw water pump. It will also act as a buffer to the engines if you come out of the water you still have some water in reserve so you don't loose pressure. They also have a removable lid letting you clean out the crap it catches and gives you a place to drop in a garden hose to run the engines when not in the water.

Did I miss anything

You forgot about keeping your intercooler clean, that is if you run one.