I am going to put myself out there by honestly saying that, as a programmer, I have never understood why people believe the above statement, and I disagree. Please don't take this as a negative comment (I know people will, which is why most people would not write a comment like this). I am just trying to understand.
I'm sorry, but I don't believe its worth its weight in gold. Here is the starting point for my perspective:
1. An analyst, customer, boss, or someone who is not a programmer, describes in words what they want the software to do. I take that description and spend a couple of days building a prototype web application. I don't create the screens in Photoshop first because A) it will take longer and B) I'm not trying to create the ultimate user experience, I am trying to build functional software that we can start getting feedback and iterating on.
2. Analyst, customer, boss, designer, or someone who is not a programmer, looks at the prototype, may or may not acknowledge that the software now has the primary functionality requested, and starts criticizing the appearance or user experience. Its actually easy for him to find flaws: its not very attractive (there was no time invested in making it attractive) and some use cases could involve fewer clicks.
2A. Here is the issue I have: it really seems to me that a significant portion of people believe at that point that the programmer was unable to conceive of an attractive design or shortcuts that would improve the user experience, and that is the context and justification for the entire field of design.
Anyway here is my experience: if you can't code, then you probably cannot actually create a better user experience than me. If I am given time to work on a design, then I will optimize the user experience as much as possible given the features of the software platform and budget. If there really aren't budget or time constraints then I can disregard the inner workings of the platform to some degree because I know I will have time to code whatever interface I can conceive of.
Maybe some designers can help me understand.