Why did the capital punishment come back?

Answer 1

It never went away. 37 states plus the federal government and US Military still have the ability to impose the death penalty.

Another view: The death penalty was effectively suspended in 1972 by the Supreme Court in Furman v. Georgia (408 U.S. 238). The court found that Georgia's statute (and by extension, most other state statutes) allowed the death penalty to be imposed arbitrarily, and that it therefore violated the 8th amendment. The Supreme Court never found that the death penalty itself was unconstitutional. Following that decision, many states rewrote their statutes, effectively "bringing back" the death penalty in most states. It was done so primarily because the people (or the legislatures) wanted to.