I can read Russian, and understand mostly what is written here:
But what I fail to see is why this is a joke? What makes it funny? I am interested what am I missing in understanding in this joke that would make it funny?
Like why is the name of cheese: Bri funny?
Or name of the champagne: briut, funny?
This is a rhetorical device known as alliteration.
Бри means "brie", a type of French cheese. Брют means "brut", a type of champagne. These words are not funny in and of themselves, but they do make the alliteration work.
Being funny is a very subjective thing. This particular alliteration is not very funny to me personally, but I can see how it can be to other people.
Alliteration is fun when you try to read it out loud quickly. You're sure to mix up and mess up the words, and that's funny. We have lots of tongue-twisters based on this principle.
Шла Саша по шоссе и сосала сушку.
На дворе трава, на траве дрова.
Шит колпак не по-колпаковски. Надо колпак переколпаковать, перевыколпаковать.
Try to say them quickly and you'll at least smile.
