Where and how do I go about hiding the posts that match any of the tags on my ignored list?
It will autosave the preference once you check/uncheck. Now, when you're viewing a list of questions, those marked with any of your ignored tags will be removed from view.
They're taken out of your view when they get to your side of things, so keep that in mind when paging through with set amounts per page listings. i.e. if you're thumbing through 50 a page, but you only count 40 on the page at a time, then 10 are being hidden.