Learn English – What does “pay you out” mean

From Wikipedia:

Corsage refers to a bouquet of flowers worn on a woman's dress or worn around her wrist.

A corsage is an item typically worn at a prom or similar event.

The use of this expression is because of the topic of legislators having to pick partners, which can draw a humorous comparison to a prom. The State of the Union address had traditionally been a time of strong partisanship. The idea to pick partners from "across the aisle" is an attempt to show the supposed bipartisanship of the current legislature.

An article explaining the details of this plan is available here. Quoting from the article:

In a letter to his fellow lawmakers, Udall said the partisan seating arrangement has become a negative symbol of the divisions in Congress - and among the American people - with one side of the chamber cheering and applauding loudly throughout the President's speech, while the other often sits silent. He urged them to bridge the partisan divide by sitting together as a symbolic gesture signifying unity and better reflecting the communities they represent.

The term "break out" is often used colloquially for "getting out" or "taking out" something. For example, "Let's break out the champagne" could be used to express the appropriateness of celebrating some event (although this particular term is often used sarcastically). A search on Google reveals some interesting use of this syntax.

YourDictionary.com has a definition as well:

1. Develop suddenly and forcefully. For example, A fire broke out last night, or He broke out in a sweat. [a.d. 1000] 2. Be affected with a skin eruption, such as a rash or boils, as in A teenager's face often breaks out in pimples. [c. 1300] 3. Prepare something for consumption, action, or use, as in Let's break out the champagne, or It's such a fine day—let's break out the fishing rods. [Early 1800s] 4. break out of. Force out by breaking; also, escape from confinement. For example, The hurricane broke the glass out of all the windows, or He broke out of prison but was soon apprehended. [Early 1600s] 5. Isolate a portion of a body of data, as in Please break out the sales figures from the quarterly report. [Mid-1900s]

No, it's actually kind of backwards from what you suggest, but you are right that it has to do with fame or popularity of something.

Specifically, the phrase is a reference to trends. When something is "trending down" or "losing popularity," it is said to be "on its way out." If it is a product, maybe it is still available in stores, but it isn't the most popular or talked about item and is losing momentum against some new trend.