There are plenty of websites talking about it but I think this one explains it well:
What Exactly Is Freedom of Speech and How Does It Apply to the Internet?
Let me quote some parts so you won't have to read everything if you don't want to.
ZitatThe truth is, what your freedom of speech is and where it applies is actually very simple, but our understanding of it has been distorted, mostly because people use "free speech" as a weapon where it doesn't apply, and often as a shield to hide behind when they're being criticized.
Zitat
Today, most of us turn to the internet because the tools are free and available: Twitter, Facebook, Reddit, newspapers, blogs with comment sections, forums, they all offer one-click methods for us to speak our minds. However, when you leave a comment on a company's Facebook page, post to a Reddit thread, or tweet your grievances, you're speaking in privately owned spaces. This means you should have no expectation that your speech is somehow protected beyond that service's terms of use.
ZitatMost spaces on the internet are privately owned, and have no obligation to allow you to speak freely in their space.
Whether it's Facebook removing content that violates its own terms of service, a blog owner deleting a comment they find offensive, or a big company deleting user posts from its Facebook page, your speech may be censored, but you have no first amendment right to free speech in those places. This includes our discussions on Lifehacker—we've always held our community up to high standards, and if you start a discussion we find isn't up to those standards, we reserve our right to dismiss it.
Zitat
While "censorship" can apply to any type of speech suppression, censorship in the context of "free speech" is generally reserved for speech that's suppressed by government or state actors. A company deleting your post on their Facebook page about how their product was a waste of your money, even if you think the post was relevant or witty, is not state suppression of your speech. It may be censorship, but your freedom of speech has not been violated.
The fact is, you can still express your freedom of speech. Just not here, if the owners of this board decide (for the reason they want), that you aren't allowed to say what you said.
It is not your choice, and freedom of speech completely doesn't apply on a privately owned forum.
Go tell Facebook they violated your rights by removing a post of yours because they considered it violated their TOS, and take them to court. I'm sure you'll win (for those who didn't see it, that's sarcasm).
ZitatSo where can you speak freely if you find your opinions are no longer welcome on acompany's Facebook page, or if you've been banned from commenting on a blog? You can set up your own space to speak—as in start your own blog and invite people to come and participate in a truly free discussion with you, or start your own Facebook page where you air your grievances with the company that won't respond to you anymore. Again, because these are actually private spaces, your speech is only as protected as your service provider's whim. If you start a free blog at Wordpress or Tumblr, you have to stay inside the lines of acceptable use. If you start a Facebook page, you have to adhere to Facebook's TOS. Make sure to read them carefully.
Your best bet is to get a decent web host and host your own blog where you can say what you like. When you host your own speech, you're now the party with the terms—you can be as lenient or restrictive as you choose. You may not have the audience you wanted (that Facebook page's subscribers or forum's readers), but free speech protects your right to speak—it doesn't force others to listen. You're still subject to your hosting company's TOS, and it's still a semi-private space, but it's the closest you can get to public on the internet, and most hosting companies don't care what you say as long as it doesn't get them into legal trouble.
So basically, as we consider a lot of your posts to be toxic for our community, we're asking you to change your behavior, or leave, or get banned.
We don't even have to justify and give you reasons for that actually. We're just nice enough to do it. But now we've told you so many times to change your bahavior that I can't remember how many times it was. So I think everyone will agree that this is your last warning.
I know you can be nice, so just be cool with us if you wish to stay here with us, instead of creating too often a bad atmosphere on these forums.
(Once again, idc if you think you're actually not creating a bad atmosphere on these forums. I know you are, and the owners know you are, and that's what counts).
Regards.
PS: For those who don't know, this post is directed to #removed, I just noticed I didn't write his name.
Edit:
Just thought of a nice example to sum it all up
You buy a house, it doesn't only give you the right to live in the it, it also gives you the right to kick out of it anyone who enters in it without your permission, or anyone who displeases you or whatever. And you obviously don't need any reason for kicking him out, because it's your property and you do whatever you want with it.
In this case, the board is the property of the people who are paying for the server it is hosted on.
They could shut it down if they wanted to and no one else would have his word to say about it.