How is it that SpaceX can do it so much more cheaply?
In addition to what was answered before, there are also several other factors keeping the costs down.
1) Production isn't only done almost exclusively within the company, it's also mostly done at the same place, reducing needs of infrastructure by quite a bit. Where they do buy parts, they often buy off-the-shelves materials and the like, and is willing to eat a little extra mass for a big reduction in expenses.
2) SpaceX has a very high degree of standardisation when it comes to what parts are used in their designs, (such as e.g. using essentially the same engine for the first and the second stage), resulting in savings because of the economy of scale.
3) They have a completely excellent staff which is incredibly motivated and works insane hours, yet they pay them relatively little. This is due to the number of highly qualified people who are willing to take less-paid positions to be able to be a part of a company that is currently a driving force in the buisness. It also looks good on the resume. Some of this motivation among employees is exactly due to the company's lofty, and possibly uneconomic and unrealistic goals of Mars colonization. It's either a company with a strong mars cult inside of it, or basically a mars cult masqerading as a company.
4) There are speculations that they aren't actually earning much of a profit on their launches, and that their buisness model is based on undercutting the competitors to expand their market share, while depending on fresh investments to stay solvent. Much like the typical surging social media company, really. I'm not sure if I buy this, but I think it's reasonable to assume their profit after expenses is lower than their competitors.
5) To be perfectly fair, they do have a higher number of rapid unscheduled dissasemblies than the incredibly expensive ULA. And that's a pretty big deal.
Lastly, they are pretty agressively trying to reduce the prices even further, with their reusable first stage program. The landings are promising, and if they turn out to be able to refurbish the used stage relatively inexpensively, we might see them selling launches for as little as 40-45 million a piece within a few years. Plus, when they get the Falcon Heavy going, that will even further reduce the price of cargo per lbs if it goes for as little as 90 (as currently advertised).
I'm excited. A more economic and accessible way to orbit is paramount to the future of space exploration.