Why should I care about that?
Freedom 0, the freedom to run the program for any purpose, is the easiest to explain. A word processor, for example, should not come with restrictions that say you can't use it to write Republican campaign materials. Because running a program on a computer to do something one wants to do is the minimal definition of what it means to use a computer at all, the importance of this freedom is easy to explain.
Likewise, Freedom 2, the freedom to redistribute copies so you can help your neighbor, is straightforward. Like freedom 0, it is not hard for everyone who encounters computers to see how this freedom directly impacts their lives. When using a computer, it is prima facie beneficial and empowering to be able to exchange with others copies of programs that do useful things.
So far, so good. But now we are left with the two freedoms for which access to the "source code" is a precondition: Freedom 1, the freedom to study how the program works and to change it to make it do what you wish; and Freedom 3, the freedom to subsequently distribute copies of your modified versions to others.
The importance of these freedoms has always been difficult to explain to people who aren't familiar with how programs are written. We lack an equivalent of Schoolhouse Rock's "How a Bill Becomes a Law" showing how source code becomes a running program. Even if we had one, we couldn't expect everyone to be interested.
Unfortunately, things on this front have recently taken a turn for the worse. As of this writing, the top result when searching Google for "source code" is not a definition talking about the human-readable language in which programs are written. It's a Jake Gyllenhaal movie. I haven't seen this movie, but I'm told that in it, the "Source Code" is a government experiment revolving around a program that enables one person to cross over into another's identity in the last eight minutes of life. I've mostly gotten over the theory that this movie is in fact a conspiracy designed to make our jobs harder. (But if you have evidence, please contact me.)
The prominence of Gyllenhaal's Source Code is just more buckshot for naysayers who claim we will never be able to convince people on a large scale that the freedom to modify a program is critical, because most people in their lives will never want to modify a program. So why should they care?
Richard Stallman has been asked and has answered this question many times. In his answers, he highlights the importance of what amounts to literacy. He says that, with access to source code, anyone can learn to be a programmer and come to value this freedom; and it doesn't take much work to learn how to do a lot, even for people who don't ever intend on becoming master programmers.
I can testify that his answer worked for me. What little I know about programming is self-taught from free software, other than a couple high school classes in Pascal and BASIC. But I know enough -- most importantly enough about how to look up answers for the things I don't know -- that I can accomplish a great deal in my personal computing environment.
Even such minimal literacy is emancipatory; it is the surest way to make sure that you are not a prisoner of the software you use. But I also recognize that this appeal doesn't work for everyone, and that's okay. I believe we can also earn the support of people who can think of nothing they would like to do less than learn or change how their computer works.
In this effort, we are up against powerful marketing machines like Apple, who spends millions of dollars to tell us repeatedly that nobody should want anything more than for their technology to just work, in the process paradoxically defining freedom to mean the antithesis of both the desire and the ability to change anything about one's circumstances. This marketing strategy runs from their software down to their physical devices, which are known for prohibiting simple things like changing the battery or installing a new storage card.
Of course people want their technology to work. But Apple's trendsetting overemphasis on this idea has made them the Fox News of technology. Just as Fox News rides on the motto "Fair and Balanced," heavy-handedly implying that viewers don't need any other news sources, so Apple and its budding copycats ride on the idea that users don't need real freedom, because they are already having the best possible experience.
Free software certainly isn't always best when it comes to functionality or ease of use, and in many places it can learn from proprietary software. But just as plenty of people who do not write news articles themselves still strongly support freedom of the press so that others can write articles for them to read, so plenty of people can come to recognize the ways in which proprietary software company efforts to crush the creativity and intelligence of those who do want to modify source code -- of the non-Gyllenhaal variety -- end up hurting everyone who ever needs to run a program to do anything. Like the news media in places with no free press, proprietary software means gate-kept control over information, tools, and knowledge, and that's scary.
"Fair and Balanced" exists in the context of the First Amendment. Likewise, companies like Apple can exist and offer their users streamlined choice-free experiences serving media certified by Steve Jobs, but they must do so in a free world, eschewing the subsidized backing of government coercion they currently enjoy in the form of copyrights, patents, and Digital Restrictions Management. They can have their image, but they must not rely on fundamental user freedom being illegal in order to do so.
Many people can and do value freedoms that they themselves do not exercise, because they empathize with the importance of those freedoms to others, and recognize that the alternative is for their own, more dear freedoms to someday be threatened ("First they came for the..."). Many people also care about the way in which the things they use are produced, and they want those methods to be ethical. When inviting new voices to the free software movement, we can and should appeal to these empathies, but we don't need to depend on them. The character of the software available to any of us is dependent on the ethical environment in which that software is made. Whether programmers or not, we all need the programs we use to be made and run in freedom.