Why are the XIII, XIV and XV amendments important and what did they do?

The 13th, 14th, and 15th Amendments are known as the Reconstruction Amendments. The United States Constitution was amended to address specific issues of law that former slave states tried to skirt around with discriminatory practices, regulations, and government enforcement. Reconstruction was a turbulent period in American history, and although the Civil War ended, the slave states did not want to relinquish their source of free and cheap labor or recognize the civil rights of former slaves.
The Thirteenth Amendment abolished slavery in the United States and territories of the United States. The amendment addressed the issue disguised as a labor practice, but in practice referred to slavery: the practice of peonage or involuntary servitude. Former slave owners sought to recoup losses from the Civil War by charging the expense of war to former slaves with the former slave having to exchange labor as payment. The practice was known as "debt slavery."
The Fourteenth Amendment was passed in response to southern legislatures passing laws that were known as Black Codes. Like peonage, Black Codes were an effort to retain the vestiges of slavery by legalizing certain practices that prohibited former slaves from seeking job opportunities. The Fourteenth Amendment is also a current topic of discussion, as it is the language in this amendment that states, "All citizens born or naturalized in the United States" are citizens of the United States. The issue is particularly salient in the debate over immigration.
The "due process clause" and "equal protection under the laws" are also part of the Fourteenth Amendment. These clauses form the cornerstone of the American legal system, and though initially aimed to protect the rights of former slaves, these provisions benefit every American citizen. There is also the language that establishes how representatives in the United States Congressional House districts are allocated. Again, with the Census a few short months away, this is an incredibly important clause.
The Fifteenth Amendment was aimed at addressing suffrage rights, particularly for former slaves. Though the Fifteenth Amendment extended the right to vote to all United States citizens, regardless of race, actual suffrage for African Americans remained very limited for decades. Many states passed laws that attempted to circumvent the Fifteenth Amendment by preventing African Americans from exercising their right to vote, including requiring that voters pay a poll tax, pass a literacy test, or own property in order to vote.
The Reconstruction Amendments (Thirteen, Fourteen, and Fifteen) were one of the first significant legislative actions in the area of civil rights. It would take nearly another century before the next significant action of Congress in the field of civil rights would pass; the Voting Rights Act of 1965.
https://constitutioncenter.org/learn/educational-resources/historical-documents/the-reconstruction-amendments

https://www.pbs.org/tpt/slavery-by-another-name/themes/reconstruction-amendments/