Why were people interested in building a transcontinental railroad?

 Speed. You have to remember that Canada, the US, and Mexico all span an entire continent, not many countries can make such a claim. Australia of course can, but it can also lay claim to being a continent unto itself. Now keeping that in mind, let's say it's the 1850s and you want to travel from Baltimore to San Fransisco. That's a trip of about 2825 miles and you've only really got four choices. 

1. Take a ship down around Cape Horn and then up to San Fransisco, the longest choice in terms of miles traveld but also the safest. 

2. By foot, which will leave you quite exhausted once you get there and will also expose you to many different dangers. 

3. By beast of burden, either riding them or in a wagon drawn by them, leaves you just as exposed as by foot to danger but may prove a bit faster. However you've got to consider how you're going to feed the horse/horses, mules, or oxen in addition to yourself as you may come across areas where there's nothing for them to graze on. 

4. Take a train as far west as you can then walk or ride to the next train stop. Keep doing so if you have to. 

Of those four, the train is going to be the fastest but it's the 1850s and as pointed out eventually you reach a point where you have to resort to a slower method of travel. So folks wanted something faster, and a virtually non-stop train from the east coast to the west coast would be it. Not only could folks travel from coast to coast faster, but they could recieve and send mail back and forth faster and trade goods would move even quicker. This would be especially helpful for perishable items. 

The transcontinental railroad meant that for the first time travel between the coasts could be cut down to just a few short days rather than weeks or months. So when the east and west railroads finally linked up in 1869 for the first time in US history it was possible to travel across the continent quickly