Historically, throughout Western Europe, Sunday was the first day of the week. This goes back all the way to the Hebrew Bible, in which the seven-day week ran from Sunday ("the first day") to Saturday (the Sabbath). It's also preserved in Portuguese, where Monday is Segunda-feira and Friday is Sexta-feira. (This contrasts with most of Eastern Europe, where there is a long tradition of the week starting on Monday).

So the question is not so much why the US considers the week to start on Sunday, but why many Western Europeans have changed to considering it to start on Monday. I imagine that it something to do with secularization and the two-day weekend, but don't know for sure; hopefully other contributors will have better sources.

Source: Zerubavel, The Seven Day Circle: The History and Meaning of the Week

There are two explanations for this really, and they both have to do with religious tradition. In the Jewish faith, Saturday is the seventh day, the one where god rested after creating everything. This would make Sunday the 'first day', so that would go first. The other reason is to start the week celebrating the resting day, in accordance with Genesis.

So the sort answer is to celebrate the Sabbath, but the deeper reasoning depending on the religion.

http://www.gotquestions.org/Saturday-Sunday.html This goes more in detail about the Sabbath and on what day it falls, with links to Biblical passages relating to the day that you should celebrate.