The man who wrote the Kansas-Nebraska Act was Stephen Douglas. What was his purpose in authoring such a law?

Answer: Senator Stephen Douglas proposed the bill that became the Kansas-Nebraska Act as a way of getting southern support for Nebraska statehood. Douglas was seeking to bring Nebraska into the Union in order to bring those lands under government authority and lay the groundwork for building a Midwestern route of transcontinental railroad that would run to Chicago and benefit his state (Illinois). The compromise to gain support from the South was to create two states, Nebraska and Kansas, and allow voters in those areas to choose whether they'd be slave or free. The thought was that Kansas might end up as a slave state and Nebraska as a free state, thus maintaining the balance between free and slave states.

Further detail:

The Kansas-Nebraska Act was enacted by Congress in 1854. It granted popular sovereignty to the people in the Kansas and Nebraska territories, letting them decide whether they'd allow slavery. In essence, this made the Kansas-Nebraska act a repeal of the Missouri Compromise of 1820, which had said there would be no slavery north of latitude 36°30' except for Missouri.

After the passage of the Kansas-Nebraska Act, pro-slavery and anti-slavery settlers rushed into Kansas to try to sway the outcome of the issue, and violence between the two sides occurred. The term "bleeding Kansas" was used because of the bloodshed. Kansas and Nebraska ended up as free states, but the Kansas-Nebraska Act had allowed the possibility that slavery could become slave states.