What is the history behind Library of Congress being the world's biggest library? It is rather new because USA is a rather young country. One would expect some library in Paris or London, or maybe the Vatican library, to be bigger, simply because they have been going on much longer (and should have had more resources up until WW2 or so).
(There was no Library tag available, could someone add that?)
Let me add to the answer of Tom Au that this is essentially the question of funding. For example, Soviet Union also had a library which had a "mandate to collect everything". This was the Lenin Library in Moscow. The job was facilitated by the law which made it mandatory for every Soviet publisher (actually every printer) to send a free copy of anything published/printed to this library. So I suppose this library had everything which was published or printed in Soviet Union. It had a sample of EVERY printed item, down to matchbox labels:-)
But of course they had to pay for books published abroad, and it was inferior to the Library of Congress in foreign (non Soviet) books, because of insufficient funding in foreign currency. You can maintain the best library in the world if you have a desire and enough money.
The Library of Congress is the "Great Library" of our time. It was designed as such, because it serves the Congress, President, and other leaders of the United States.
Early on, it was "seeded" with the entire personal library of (former) President Thomas Jefferson. Thereafter, it had a mandate to acquire two copies of every book published in the United States. Even if another national library has such a mandate, their economies, and hence their publications would be smaller.
After World War II, America made great efforts to "digitize" and "internationalize" the Library of Congress, probably more than any other country in the world. Basically, the Library of Congress defines "national book" more broadly than most. And even if "other" libraries may have had more resources "up to World War II or so," it's been 75 years since then, with that period covering the "American century."
This is an interesting question because the questioner is asking why the Library of Congress is the largest Library in the world, when compared with other likely competitors, such as the Vatican Library or the British Library.
While I do not have a definitive answer to this question and the Wikipedia article on The Library of Congress is limited in its information on this topic, there is perhaps one theory that MAY help answer this question. The Library of Congress' enormous size MAY be related to the evolving diversity of the American populace over the centuries and with it, the massive volume, availability and accessibility of traditional, as well as diverse literatures, documents and other texts.
It is estimated that The Library of Congress has 170,000,000 volumes; if one were to average this, that would be a single book, text or document for 1 out of every 2 U.S. citizens. Where do these multitudes of literatures, documents and other texts come from? 200 years ago, The Library of Congress originated as the posthumous collection of Thomas Jefferson. But, as the United States grew and expanded over time, the Library of Congress' availability of topics have become more encompassing and sophisticated over the years as well.
In a way, The Library of Congress has reflected and continues to reflect, the larger cultural character of America itself; a heterogeneous and heterodox cultural character. The Vatican Library reflects a traditional Roman Catholic Christian character, the British Library, while quite diverse in its collections, is a Library which reflects the cultural character of Britons-(of various backgrounds). Yet, The Library of Congress is an evolving intellectual institution that reflects the diverse and rejuvenating cultural character of the United States.
Admittedly, this is a theoretically based answer with limited evidence to support such a position. However, it is an answer that is not so implausible or unimaginable.