The 15th Amendment was the last of the “Reconstruction Amendments" to be adopted. It was designed to prohibit discrimination against voters on the basis on race or previous condition of servitude. Previously, the states had had full responsibility for determining voter qualifications.
Reasons for supporting the amendment are not immediately evident, but they went far beyond an idealistic desire to spread the fruits of democracy to former slaves.
In the Election of 1868, Grant achieved a narrow majority of the popular votes nationwide. His support from black voters in the South made the difference. Without those votes, he would have lost. The largest state Grant lost was New York (home state of Horatio Seymour, his opponent), which was conceded by a narrow margin. Blacks could not vote in the North - if they had had that right, Grant would have taken New York.
The main impetus behind the 15th Amendment was the Republican desire to entrench its power in both the North and the South. Black votes would help accomplish that end.
The measure was passed by Congress in 1869, and was quickly ratified by the requisite three-fourths of the states in 1870. Republicans still controlled the state governments in the South, so the expected opposition lacked the means to block the amendment.
Was the 15th Amendment successful? Yes and no. It did provide the vote to blacks living in northern states, and it did encourage voting by blacks in the South for a period of time.