Please help me write an essay comparing and contrasting two Martin Luther King, Jr. speeches: "I've Been to the Mountaintop" and "Remaining Awake."

Comparing and contrasting are two important skills to understand. When comparing two things, you are finding the similarities between them. When contrasting two things, you are finding the differences between them.
The first step you should take is to find some background information on each speech, such as when and where the speech was given. This information will help you write your introduction and conclusion paragraphs.
The next step you should take before writing your essay is to carefully read and annotate each speech. When you annotate a text, you add your own notes to it. You can also give explanations, definitions, or comments on the text. I personally would make a copy of each speech and write my notes on the actual speech. This way, you can highlight, underline, and circle important information and also write notes. I would first look for similarities between the two speeches and then I would annotate again, looking for differences.
Once you have at least three similarities and three differences, you can start to write your essay.
I would begin by writing the introduction paragraph and focusing on your thesis statement. Because I am not sure of your exact prompt, I can only give some suggestions. If this is an informative essay, you would begin with background information. You would then write your thesis statement such as: Martin Luther King Jr. wrote and delivered many speeches such as his "I've Been to the Mountaintop" and his "Remaining Awake" speeches; his speeches have many similarities and many differences, but they still impact the world today.
Next, you would use your research to write your first body paragraph discussing similarities. Be sure to use specific evidence and to cite your sources.
Using a transition sentence, you will write a second body paragraph about the differences between the speeches. Again, cite your sources and use specific evidence.
For your conclusion paragraph, if this is an informative essay, be sure to summarize the main idea of your essay, restate your thesis statement, and make a connection to something else. This connection can be text-to-self, text-to-text (in your case), or text-to-world.
Hopefully this gives you an idea of where to begin. Good luck!