Question

What factors might explain the widespread support for a transcontinental railroad in the 1850s and 1860s?

Answer 1

They support the construction because Prior to this, long-distance travel was slow, and the spread of goods and information was nearly nonexistent between faraway states. 
This make it difficult for producer such as farmers to distribute their products and sell it other people outside their area.
The Development of transcontinental railroad cut down both the cost of production and the time of delivery of the farmers. On top of that, the construction also provides a lof of labor jobs for the people.

Answer 2

Factors that include support for the transcontinental railroad include trade routes, as the panama canal was not built yet. The United States was going through a period of Manifest Destiny, in which Americans believed they were entitled to the land in the west. The Gold Rush and rapidly expanding California desired to be linked to the already established east coast.