how did the radical republicans come into power?

Question

Radical Republicans supported voting rights for black men.

Answer 1

Good question. Contrary to what many think, though the "radical Republicans" were an important force from early in Lincoln's first term, the majoriity was held by the moderates.

That is, until President Johnson, despite his early rhetoric proved EXTREMELY soft toward the South, quick to grant pardons, etc., and not at all concerned about the plight of the freedman. He essentially REFUSED to work with Republican efforts on the Reconstruction INCLUDING those of the moderates. This climaxed in his vetoing the Civil Rights Act of 1866 when for the FIRST time in American history, Congress overrode a Presidential veto on a significant matter.

Johnson continued to be stubborn and to advocate his own positions and often more friendly to the Democrats' positions. But his vigorous campaigning in 1866 was a disaster -- he was widely ridiculed, and the Republicans won in a landslide. Those who most opposed Johnson (the radicals) were now in power and able to begin to push through THEIR program of Reconstruction, beginning in 1867.

Here is a further description of the Johnson - Republican falling out and its consequences -- note esp. the assessment by James Ford Rhodes on the key role Johnson's own stubbornness played in it all.
http://en.wikipedia.org/wiki/Andrew_Johnson#Break_with_the_Republicans:_1866

see also:
http://en.wikipedia.org/wiki/Radical_Republican#Reconstruction

Answer 2

By having reasonable ideals and a reasonable agenda.