In a country where government is not protecting the right of free speech, or where its protection is not enough to protect you from powerful organisations seeking to stifle it (which is pretty much every country in existence, since the government can not control everything everywhere) anonymous speech may be the only way to achieve free speech.
Anonymous speech has a lot of value, which can be obvious from the fact that so many efforts are taken to stifle it. If it were ineffectual, nobody would bother to inhibit it. So it is obviously effective in influencing people. If it is effective, it has value.
