What were the reasons given by Hitler and his followers for carrying out the Holocaust?

The answer to your question, the actual reasons given for the Holocaust, were most directly outlined at the Wannsee Conference in 1941. While much conjecture of what was said between the Nazi hierarchy exists, few written documents outlining specific goals and reasons for the Holocaust were written down, for obvious reasons.

For details about the Wannsee conference check out:

Recent academic discussion regarding the documents and proof of what is said to happened are being dismissed by many mainstream educators. This is valuable to consider at what point the facts are cleared and vetted by history, which at this stage is surprisingly young in its scope and view of the Holocaust.

Yet in Eichmann’s Trial in Jerusalem, the first specific mention of what was planned at the Wannsee Conference, where the term “Final Solution” was coined, and the specifics details (though not actual reasons) were given, Gary Grobman has a good write up here: https://remember.org/eichmann/study2.htm

“Fifteen high-ranking Nazi officials were in attendance, including Heydrich, for whom Eichmann prepared the conference’s opening speech. At this meeting, the Third Reich adopted a plan to deport Jews to death camps, in what became known as the “Final Solution.” Most of the debate centered not on the morality of this,but rather what proportion of Jewish blood would merit this “special treatment” and the methods used to effect genocide. Heydrich made it clear that all organizational and logistical requirements would be the responsibility of Adolf Eichmann.”

The actual reasons likely varied among members of the Nazi party, from anti-Semitism to simply eliminating the Jews from Germany. In fact, some historians point to evidence that Hitler tried to simply rid Germany of the Jews, sending them away to various places like Madagascar, and many countries would not accept this massive emigration.

The reasons are varied, but the only documents that outlined the actual Holocaust and how it would be executed can be found at the Wannsee Conference.