What occurred during World War I and World War II that tested the nation's commitment to the rights and civil liberties of its people? Why did these things happen?

It is important, when considering domestic and social issues in the United States during the early part of the twentieth century, to keep in mind how recent was the seminal event of the previous century: the Civil War. The Founding Fathers of the United States of America had, during the Constitutional Convention and in the years that followed, failed to resolve key differences regarding slavery and the rights of those of African heritage. Unable to resolve those differences, the civil rights can was, so to speak, repeatedly kicked down the proverbial road until the question of Southern secession forced the issue. The war, and the period of Reconstruction that followed it, was still relatively fresh in peoples' minds, while the "Great Powers of Europe" moved inexorably toward the massive conflagration that would become known as "the Great War" and, later, as "World War I."
In addition to the continued denial of basic civil rights to the nation's African Americans, the United States had continued to deny certain rights to women, including, most notably, the right to vote. So, again, as war approached across the Atlantic Ocean, and as the United States struggled to reconcile its ideals with its practices, the issue of women's suffrage loomed large. The suffragette movement was gaining in strength as the twentieth century began, and as American troops were sent overseas to fight in the Great War, the debate in Washington, DC, over suffrage intensified. The result of this debate, as we know, was ratification, on August 18, 1920, of the 19th Amendment to the Constitution, the text of which reads:

"The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any state on account of sex."

Overlapping with this development in women's rights was the "Great Migration," the mass movement of African Americans from the segregated South to the more promising locales above the Mason-Dixon Line. 
While major developments on the civil rights front were being experienced, however, the Great War allowed for the passage of laws that were considered inimical to the aforementioned ideals upon which the country was founded. The first such law was the Espionage Act of June 15, 1917. As American entry into the war was underway, and in the context of specific German activities that helped propel the United States into the war on the side of England and France (i.e., the so-called "Zimmerman Telegram" and German submarine warfare affecting shipping between the United States and Europe), Congress passed legislation making it a crime to collect or provide information "respecting the national defense" the intent or purpose of which "is to be used to the injury of the United States, or to the advantage of any foreign nation...." In other words, do not get caught photographing potentially sensitive locations or buildings and certainly do not get caught passing those photographs to nationals of foreign nations with which the United States is at war.
The other legislation passed into law was the Sedition Act of 1918, a considerably more controversial measure that made it a crime to act in a manner deemed unpatriotic. While certain of the Sedition Act's provisions simply reinforced the provisions of the Espionage Act, this latter law went well beyond the earlier legislation by criminalizing disloyal actions that should have fallen within the constitutional protections outlined in the Bill of Rights. The Sedition Act specified as criminal the intentional conveyance of statements, written or spoken, that were "disloyal, profane, scurrilous, or abusive language about the form of government of the United States or the Constitution of the United States...." In addition, the Act made it a crime to interfere in the federal government's efforts at conscripting individuals into the Armed Forces or interfering in any way with the government's actions with respect to an ongoing war, in this case, the Great War.
The Espionage and Sedition Acts both represented serious challenges to the rights of Americans, and both were, obviously, passed in the context of American entry into and participation in the war.
As what would become known as World War II approached, the United States continued to be seriously divided on the issue of civil rights. While women had secured the right to vote, discrimination in the workplace continued, and, with respect to African Americans, racial segregation and other forms of discrimination continued to exist. The Great Depression had ravaged the economy of most industrialized nations, and massive unemployment figures only exacerbated the situation for black people and women already limited in opportunities. The Second World War, however, helped move the yardstick to a certain degree. With able-bodied men being sent to Europe and Asia to fight in the war, their jobs were filled by women. For black people, however, the situation remained dismal. The United States Army was segregated, with African Americans restricted to menial assignments because they were deemed inadequate for the discipline and rigors of combat--notions that were dispelled when they were finally provided the opportunity to serve their country in combat positions. In fact, both African American and Japanese American units were eventually formed, and both served with distinction. The sons of Japanese Americans interned by President Roosevelt after the Japanese attack on Pearl Harbor were eventually allowed to serve, and the 442 Infantry Regiment, comprised almost solely of such individuals, was the most highly decorated US unit in the war, while the Tuskegee Airmen, African American fighter pilots, similarly served heroically once given the chance.
The facts of World War II made the domestic status quo untenable with respect to minorities in the Armed Forces, and, on July 26, 1948, President Harry Truman signed Executive Order 9981 officially and effectively desecrating the Armed Forces of the United States.
Back on the home front, President Roosevelt had, early in American participation in the war, signed Executive Order 8802 (June 25, 1941), which established the Committee on Fair Employment Practices. This act made it unlawful to discriminate in the workplace on the basis of race, although, as we know, far more would need to be done to eliminate racial discrimination in the United States. While Roosevelt's action was intended to address problems experienced by black people working in factories that supported the war effort, the executive order served as a model for future legal measures.
These, then, are the main civil or human rights-related developments associated with American participation in the world wars. Of particular relevance was the ratification in the Second World War's aftermath, by the newly established United Nations, of the Universal Declaration of Human Rights, the text of which reiterated the fundamental rights guaranteed to all humans. 
https://www.nationalarchives.gov.uk/humanrights/1914-1945/

https://www.ourdocuments.gov/doc.php?doc=72