I imagine this is related visa regimes and the checks airlines need to make to stop people being denied entry at a destination for not having the proper visas or documentation. If a passenger is denied entry for that reason then it's the airline who have to pay for their return to the point of departure.
If you hold an EU passport then you have the right to travel to any other EU country, in fact you can travel to any European destination with no visa with the exception of Russia, Belarus, Armenia and Azerbaijan (sometimes Turkey, depending on the flavour of your EU passport). That about covers RYR route map.
The problems arise with holders of non EU passports. For example, if you hold a Serbian passport and possess a valid visa for the UK this does not entitle you to visit Schengen. Also, you may actually have a Schengen visa but your UK visa is single entry, which means they won't let you back into the UK. The airlines are responsable for checking travellers have valid travel docs which is why RYR prefer non EU passport holders to check in with a human being rather than on the web.
Some airlines are happy to do a detailed document or visa check at the gate but it's normally supplemental