Tabletopia is currently in closed beta. The official release won't be until sometime in the first quarter of the year, and when that happens it will be available for free at the bronze level. You can pay for a month early access for $10 and it costs more if you also want a subscription or if you want earlier access. I would advise against paying for early access if you're not planning to get a paid subscription in the long run, since it's not going to be too long before it's free at the bronze level.
This question has been asked countless times and there's a lot of people who aren't properly informed about it. Even Arcane Wonders was misinformed when I asked them about it. They said that tabletopia wasn't going to be free, but they're wrong. It will be free at the bronze level once it is officially released. In the article on the AW website, Arcane Wonders made it sound like they thought that tabletopia was already officially released.
While a lot of information about tabletopia is spread out all over the place, once you know where to look it's very easy to find out the answer to this question.
Go to:
https://tabletopia.com/Click "Join Now" and then Click "Read More" and scroll down a bit. That should answer your question. There's also a timeline for tabletopia development near the bottom of the page. The information is all right there on the website.
When I first found out about tabletopia it was a wild goose chase to find this out. First I saw the article on arcane wonders about it, where it did not say how much it would cost or if it would cost anything, only said to "click here to start your first game of mage wars academy on tabletopia!" and i clicked the link and it turned out i needed an account, and that in order to create an account I needed an invite code. Then I googled it and found a tabletopia blog which didn't answer the question at all but was interesting to look at. Then finally I found the tabletopia website and went to the join now section, clicked "read more" and found out that it would be free at the bronze version once the app is officially released. I googled the release date, and according to steam it was Q4 2015, or in other words, the fourth quarter of the year. But it was already nearly the end of the fourth quarter of the year. Finally I asked on tabletopia's facebook page, and found out that it wouldn't be released until the first quarter of next year. Then later it turned out there is a timeline on tabletopia's main website, and it says there flat out that the official release won't be until 2016 so I shouldn't have even needed to ask, I could have just clicked "join now" and then clicked "read more". Unfortunately, that wasn't completely obvious at the time. If you didn't click "read more" you would never find out that there was a free level, since absolutely NO ONE is talking about the fact that it will be available for free (although if you want some extra cool features, you might want to consider getting a subscription. I think I'm going to wait until I've tried a free account first though before I decide whether to upgrade or not).
Hope that helps!