There are a huge number of modern board games to play digitally now.
For most games, there is very little hope for the publisher to make any money out of it, considering the cost of development. Only for a tiny number of games, I think, it is a viable option.
Most board game apps are not free.
On browsers however, the huge majority of conversions are free. In that case, publishers ask for less, usually nothing to license their games (one big platform actually asks for money!). Online play is used as an advertizing tool. The more people can try the game online and enjoy it, the more chances they will buy the physical game.
Usually games that are free-to-play on online platforms are less polished than apps, but they also tend to have a much longer life. With many apps, it is hardly possible to find anyone to play with after a couple of days after release.
Please check out Happy Meeple (http://www.happymeeple.com/fr/). This is the most beginner-friendly online platform. It has tutorials, good AI, no waiting time, sound/music, animations. It has fewer games than others but they are clearly more polished, easy to understand, quick to play. It is the perfect place to start with the hobby.
I hope this is useful.