There are two ways to answer this.
First, it is not technically correct to say that there is a constitutional amendment that serves only to give African Americans the right to vote. The Fifteenth Amendment forbids denying anyone the right to vote "on account of race, color, or previous condition of servitude." This protects any racial group, not just African Americans, from being discriminated against.
However, the Fifteenth Amendment was meant to protect African Americans. The reason for this is that they were the group who were denied the right to vote in a systematic way. This did not happen to poor white Southerners or to immigrants from Europe.