Alright, so Radical Reconstruction conveniently occurs the year before the Election of 1868. Southern states were just readmitted into the Union with Federal military units occupying those states. Coupled with the passage of the 13th, 14th, and 15th Amendment, black could legally vote. Prior to that, freedmen were prevented from voting by the KKK and discrimination at the polls. Federal troops curbed violence and allowed for Southern States to turn red, allowing for the election of Grant.

To answer your question, Radical Republicans didn't give up, the Northerners did. The Civil Rights Act of 1875 was overturned in the Civil Rights Cases. Redeemers took control of the South. Blacks weren't adequately protected by the KKK Acts or the Force Acts. The Slaughterhouse Cases and US vs. Reese weakened the 14th Amendment. The Panic of 1873 occurred and the American economy tanked. Northerners reexamined the cost of occupying the South.

These senators and legislators passed numerous bills and legislation, but they were either overturned or rewritten. The Radical Republicans, to answer your question, "gave up" when they agreed to withdraw troops from the South if Hayes were given the presidency rather than Tilden.

I can answer follow-up questions if you have them.

Source: Enduring Vision, Chapter 16

Edit: Fixed a word.