There's no experiment and it has nothing to do with the EU. It's a local thing that didn't initially even have central goverment's backing. Public transport in Tallinn (1/3 of Estonia's population) was made free in 2013 and its expenses are covered from personal income tax from Tallinn's residents.

As to other towns and rural regions, which are poorer, the central government recently passed a measure to increase subsidies to municipalities in an amount that would allow them to make all public transport free, if they wanted. It's now up to municipalities to find a balance between fully subsidized, partially subsidized and commercial lines.