To "say" something means to utter words to convey information, or a meaning. For example, "my mother says I should study harder so that I do well in school". This is a literal example, but you can use the word "says" in other contexts as well. If you look at a piece of art, doesn't it "say" something to you? In other words, all kinds of things convey information and meaning, even if we don't utter words.
"nothing says casual like..."
In this context the person is being sarcastic. Inviting a hundred people over is not "casual". It is a big event, and the big event "says" that the person thinks it's very important thing. But, the person is trying to pretend that it isn't a big thing. However, even though he says it isn't a big thing, the event itself says something different.