Like you, Snowden's freedom of speech line never impacted me... until I read this article. It suddenly hit me. The reason I was missing his point is because I was framing it in terms of what's in it for me rather than looking at it as what's in it for us. Someone who doesn't care about freedom of speech doesn't care because he doesn't see what's in it for him. But I doubt you'd argue the benefits of the first amendment.
Similarly, privacy is very important. You might not care (even though you really do), but defending privacy is about ensuring security. Privacy is important for all of us, just like freedom of speech is.
As for what the actual problem is, the problem for the most part is ignorance and a failure to quench it. We need more privacy / cyber-security advocates who can educate people on why they ought to care. It's like teaching people why it's important to lock their doors at night or why they should put their letters into envelopes instead of just using post cards. It's why my mom had to drill into my brain the importance of not giving out my social security number willy nilly. Are you so liberal with your SSN? You don't care about privacy, so would it bother you if Facebook or Google asked for it. After all, they just want to make sure you are who you say you are.
Things aren't obvious to us until they're obvious, and then it feels like common sense. DUH, lock your door! DUH, encrypt your messages!