I narrowed the outperformed in this question to the US and UK, but please feel free to include other First World nations' universities (eg: Canada, France).
[Source:] The top American universities didn't really compare to the best in Germany, France, or the UK before then.
This subsequent question inducted this post.
Prior to the 1930s, Germany had some of the world's oldest and best universities. This is largely a due to historical reasons discussed in this article
But what changed in the 1930s? Rather than ask why German Universities out performed US universities before the 1930s, perhaps a better question would be "Why did US universities out perform German universities after the 1930s?"
I would answer that by saying the two events caused the change. First, the influx of refuge scientists prior to World War II boosted US Science (e.g., this article). Probably the most famous scientist to flee Germany to the US was Albert Einstein.
Second, the US funded science much better than Germany after World War II. This extra funding was partially done out of fear of Soviet science and partially an artifact that the US economy was the only one not destroyed or heavily damaged by the war. (This article talks about US Science funding and provides more insight into this specific topic)
Also during the post-war period, many scientists choose to come to the US because of research opportunities. As described in the book Moon Shot, Wernher von Braun chose to surrender to the US because he knew only the US and Russia would have the resources for a space program (he surrender was part of Operation Paperclip). He chose the US because he would rather live here.
Similarly, many scientist immigrants continue to chose the US because of opportunities not available in their homelands. The NSF describes this here.
I don't know whether you're asking "How did they do it?" or "How do we know they were better?", but I will try to answer the second question.
I went to the Nobel website sorted by affiliated university and counted the countries and years there (no guarantee that I didn't miscount or something). I chose this source and not the Wikipedia list, because on the Nobel page, each researcher is only once and only Physics, Chemistry, Economy, and Medicine are counted.
Top Nobel-Prize winning Nations 1901-1930: Germany (28), UK (16), France (15), USA, Netherlands, Sweden (6 each).
Top Nobel-Prize winning Nations 1931 - 1960: USA (61), UK (23), Germany (22), Russia (6).
Top Nobel-Prize winning Nations 1961 - 1990: USA (128), UK (31), Germany (13), Switzerland (11).
Top Nobel-Prize winning Nations 1991 - 2015: USA (166), UK (20), France, Japan (14 each), Germany (12).
In this very limited view of research history, Germany seems to have been more prolific in research in the first third of the 20th century than the other countries.
Since the second third of the 20th century, the USA is the clear leader.
Because the linked question asked about the top universities, here they are:
Top Nobel-Prize winning Universities 1901-1930: Sorbonne University, Paris, France (6), Goettingen University, Göttingen, Germany (5), University of Cambridge, Cambridge, United Kingdom (4), Berlin University, Berlin, Germany (4), Munich University, Munich, Germany (4).
Top Nobel-Prize winning Universities 1931 - 1960: University of California, Berkeley, CA, USA (7), Columbia University, New York, NY, USA (7), Harvard University, Cambridge, MA, USA (5), University of Oxford, Oxford, United Kingdom (5), University of Cambridge, Cambridge, United Kingdom (4), California Institute of Technology (Caltech), Pasadena, CA, USA (4).
Top Nobel-Prize winning Universities 1961 - 1990: Harvard University, Cambridge, MA, USA (16), Massachusetts Institute of Technology (MIT), Cambridge, MA, USA (10), University of Cambridge, Cambridge, United Kingdom (7), California Institute of Technology (Caltech), Pasadena, CA, USA (7), Stanford University, Stanford, CA, USA (7), University of Chicago, Chicago, IL, USA (7), Rockefeller University, New York, NY, USA (7).
Top Nobel-Prize winning Universities 1991 - 2015: Princeton University, Princeton, NJ, USA (9), Stanford University, Stanford, CA, USA (8), University of Chicago, Chicago, IL, USA (8), Columbia University, New York, NY, USA (8), Massachusetts Institute of Technology (MIT), Cambridge, MA, USA (7), University of California, Berkeley, CA, USA (7).
Warning, pure speculation ahead!
My theory is that the possibility to travel played a role in this. At the beginning of the 20th century, we have Europe with its long research tradition and many established universities on one side of the Atlantic ocean and the USA with comparatively young universities on the other side. Crossing usually took weeks on a ship, so it was a bit more difficult for the american universities to hire great scientists and to participate in academic exchange.
This started to change when the first scheduled trans-atlantic passenger flights were offered in 1931. Nowadays, we can collaborate with any researcher in the world via skype, meet at conferences all across the globe, and moving across the atlantic for a postdoc doesn't mean that you won't get to see your family again for centuries.
For some less speculative reasons why the US outperformed Germany after 1930, see Richard Ericksons answer.
There was, and still is, a strong anti-intellectual current in U.S. socio-economic and cultural life, in contrast to the cultures of UK, France, and Germany (for example). The pretense is "practicality", so that (in my own experience) John Purdue, one of the original donors of Purdue University in Indiana, specifically prohibited the teaching of Latin or Greek, and playing of string instruments (as opposed to "band music", which in English means woodwinds and percussion). What I recall from childhood is an exaggerated emphasis on practicality, and a belief that everything can be made fairly simple: not really "simple", perhaps, but that all the crazy "European science" (e.g., quantum physics) was "making things more complicated than they needed to be"... since, surely, some good measurements in experiments would do better.
In the mid 20th century, too, in mathematics, there was a backlash against the "fancy" European mathematics, manifest in many textbooks, which I will not name.
So, yes, in some regards the post-WWII (and atom bomb) era, and the post-Sputnik era, and the post-emigre era, were a very different phase for U.S. academic mathematics (and other basic science). E.g., earlier, J.J. Sylvester tried to get a math program moving at Johns Hopkins, but was mostly frustrated, and departed. There was simply no niche in U.S. culture then (or now, actually) which would allow it to make sense...
My own perception of the current "boom" is that it is not really a boom any more, since things have mostly collapsed back to a hyper-utilitarian scenario... easy to justify to constituencies who are themselves inadvertently short-term-hyper-utilitarian, or otherwise ... compromised.
