Proof reading business letters is important because you do not want to send a letter to a potential business partner or to a customer if the letter has stupid mistakes in it. If you send a letter like that, you will be less likely to get what you want from the person to whom you are sending it.
If you send someone a business letter, chances are you want to make a good impression on them. You want to make them think well of you. If you make mistakes of the sort that could be caught by proof reading, you are much less likely to accomplish this. You will make them think that you are incompetent or inattentive. Neither will help you.
You might think that you can simply rely on spellcheck to catch your mistakes. However, this is not the case. You might misspell a word in a way that makes another word that you did not intend. You might write “their” when you mean “there.” You might have made a mistake in typing the name of the person with whom you are corresponding. Spellcheck will not catch any of these mistakes.
If you do not proof read a business letter, you risk making dumb mistakes that will make you look bad and cost you business.