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:
Develop suddenly and forcefully. For example, A fire broke out last
night, or He broke out in a sweat.
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]
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]
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
Isolate a portion of a body of data, as in Please break out the sales
figures from the quarterly report.