UI is important because it affects the feelings, the emotions, and the mood of your users. If the UI is wrong and the user feels like they can't control your software, they literally won't be happy and they'll blame it on your software. If the UI is smart and things work the way the user expected them to work, they will be cheerful as they manage to accomplish small goals. Hey! I ripped a CD! It just worked! Nice software! Wooooooooooo!
To make people happy, you have to let them feel like they are in control of their environment. To do this, you need to correctly interpret their actions. The interface needs to behave in the way they are expecting it to behave.
Thus, the cardinal axiom of all user interface design:
A user interface is well-designed when the program behaves exactly how the user thought it would.