"Drop and give me twenty" is a pop cultural reference from the movie "Animal House" (1978).
The poem you reference is written by humorist Andy Borowitz, who is most certainly not a Paul Ryan supporter. Borowitz makes multiple references to popular culture, and particularly popular culture that is less popular in Manhattan than it is in the Midwest (and, obviously, Wisconsin), including various comic book heros, movies, and sports. Overall, the writer's intent is to ridicule Ryan as a macho, overly-juvenile, simplistic man. You may determine for yourself if you think his creation has achieved that goal.
Animal House has many "memorable quotes" that are in common use in America including the one you ask about here: "Drop and give me twenty": http://www.imdb.com/title/tt0077975/quotes?qt=qt0479948
The character speaking this quote, Doug Niedermeyer, heads the campus ROTC and is the main foil to the movie's heros. He is a "square", militant, angry, and enjoys asserting his authority over others. "Drop and give me twenty" is a command for an inferior in his ROTC squadron to do twenty push-ups, and indirectly, a means by which the movie shows Niedermeyer flaunting his higher status and achieving satisfaction from it.
In the poem's context, this movie quote alludes to Ryan's penchant for pre-dawn workouts and the politically liberal magazine's perception of Ryan as a frat-boy / empathy-less militant. By drawing a connection between Niedermeyer, whom the audience does not like, and Ryan, Borowitz hopes to have the negative perceptions transfer to his political nemesis.
We will see much more writing of this nature over the next two months!
[BTW, the Niedermeyer character, and the above quote, re-appear in the 1984 teen rebellion anthem "We're Not Gonna Take It" by Twisted Sister, with Niedermeyer representing the unempathetic adult authority against which we all must struggle. Why, you may ask, do several large men dressed in awful drag, playing guitars far too loudly, perfectly express teenage angst? I probably had a good answer for you thirty years ago, but for right now I'm too busy enjoying the song :)
The quote is at 4:14 in this YouTube video: http://www.youtube.com/watch?v=4xmckWVPRaI ]
A sweeping statement is a statement which covers a broad topic in a concise or even terse manner. It's a simile: a sweeping motion covering a broad area in one simple movement.
Often, it is used in a pejorative sense to declare that the speaker has over-simplified a topic, or has made assumptions which are incorrect. In other words, it is sometimes used to mean that the speaker has covered a topic too broadly. The term sweeping generalisation is common.
In your example, the term 'sweeping phrases' is indeed referring to the sentence in the speech marks, as you guessed. However, it is not being used to assert that the speaker is either wrong or right, it is simply saying that the statement he makes is very broad in scope. Mr. Hidalgo is referring to a specific set of information and, in one short sentence, makes two claims about the entire history of the human race! A pretty broad statement by any measure and, given that it is offered so concisely and without any justification, it is a sweeping one.
You could paraphrase it like so:
"as such extremely broad yet concise sentences make clear..."
Your second question is related to your first. Adding to something is synonymous with contributing to it. See these definitions and examples from a dictionary:
As you can see from the above examples, you do not therefore need a specific object with the verb adding; subtitute the equivalent contributing and that will make it clearer.
Therefore, what the paragraph means in total is that Mr. Hidalgo believes that his area of research is relevant to humanity as a whole, not just to his academic discipline. So, in pursuing that research he is not only contributing to his specific field but also to contributing to our understanding of our entire race.
Best Answer
What you're observing is a phenonmenon called lexicalization. Consider the following sentence:
Here the speaker assumes that some problem exists and asks what that problem is. "The matter" means "the matter of concern"; i.e., the thing that we should be worried about. This makes perfect sense on every level of analysis.
But through the repeated use of this expression or some similar one, the phrase "the matter" has become lexicalized into an expression of its own, carrying the sense of "something that we should be worried about". The word "the" here cannot be analyzed on its own. That would be like if you said that you were going to say something, and I asked you where you were going. Here the word "going" is relieved of its usual semantic duties in order to work a side gig as part of a lexicalized aspect marker.
Is this lexicalized usage of "the matter" logical? Not especially, but langauge doesn't need to be logical—not even formal, educated, standard language. If it intelligibly communicates its intended meaning, then, by definition, it makes sense.