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.