The Watsons Go to Birmingham—1963 takes place right smack dab in the middle of the Civil Rights Movement, so you know people are going to being thinking a lot about race. Here's what was going down around then: changes were being made to how society treated African Americans (awesome) and many people who weren't comfortable with those changes reacted with hate and violence (opposite of awesome). There were some very negative attitudes floating around out there about the African American race as a whole, and the violence was often directed at anyone and everyone who was black—even, as we see at the end of the book, little girls attending Sunday school. The Watsons have to face these issues head on when they land in Birmingham, and they'll never be the same.
Being African American doesn't have any effect on the Watsons at all until they travel into the South.
Being African American is an important part of who the Watsons are. No part of the story would be the same if they were a different race.