What is King referring to in this quote from his "I Have a Dream" speech? "In a sense we've come to our nation's capital to cash a check."

In order to understand what Martin Luther King, Jr. is saying with this quote from his “I Have a Dream” speech, you will need to read the rest of the paragraph that starts with the quote.  Basically, King is saying that the African Americans in the March on Washington are there to get the rights that the government owes them.

When someone writes you a check, they promise to pay you the amount of money...

In order to understand what Martin Luther King, Jr. is saying with this quote from his “I Have a Dream” speech, you will need to read the rest of the paragraph that starts with the quote.  Basically, King is saying that the African Americans in the March on Washington are there to get the rights that the government owes them.

When someone writes you a check, they promise to pay you the amount of money that is indicated on the check.  When you cash the check, you get the money that they have promised to pay you.  When King refers to checks in his speech, he is saying that the United States has promised something to African Americans.  He is saying that they have come to cash that check, meaning that they have come to get what the US promised to give them.

So what was it that the US promised African Americans?  To find this out, look a little further in the paragraph that begins with the quote you mention in this question.  In the middle of the paragraph, King says that, when the Declaration of Independence and the Constitution were written, the US promised

that all men, yes, black men as well as white men, would be guaranteed the "unalienable Rights" of "Life, Liberty and the pursuit of Happiness."

As King says, it was obvious in 1963 that the United States had not come through on this promise.  Segregation and racial discrimination made it so that African Americans were certainly not guaranteed these rights and were certainly not being treated as if they were created equal.  By coming to Washington in 1963, African Americans were demanding that the US live up to its promise.  They were saying that it was time for the US to keep its promise and give them the equal rights that it had said all Americans should have in the Declaration of Independence and the Constitution.  This is what King means when he says that the marchers have come to Washington to cash a check.