Because in the 1800's somebody figured out that since the President has to go around campaigning across the country, he winds up becoming a repository of people's expectations for the coming years. This isn't strictly true, but it makes a good angle to go around saying you'll argue for the people's behalf.
Basically, the President, as a singular, national figure always has everyone's attention. He can make pronouncements and challenges, to which individual people then respond to their own representatives who actually craft legislation. The infamous Bully Pulpit - if enough people support what you have to say, the people who actually make laws find it in their interest to listen to you too, or disagree with you at their peril. Representatives are always thinking about the next election, and saying that you "helped" or "blocked" a President's ideas (based on how your district thinks of him) is the defacto method of campaigning. Essentially, he's a rubric, from which everyone has to take their queues.
Plus, the veto power means that every Congressional action has to either meet the President's approval, or be so popular among a mixed house that it can override anyway. The veto doesn't see a lot of use, because Congressmen don't go through the rigmarole of testing it. Instead, they go talk the President to negotiate what he'll sign off on, then make a bill that fits that.
Look, I get that you're reserved about Obama or whatever, but he didn't give the office the unofficial power it has. Vast armatures of American political power have just evolved out of convention, with no specific basis in policy.