I can think I can shed some light on Italy(one of the biggest E.U members to have this high percentage of Euroscepticism in this graph.
There has been an huge surge of euroscpeticism in Italy after Italy has been basically left alone in dealing with migrant crysis(at least that's how it's percieved here). And it has not been a thing relegated to fringe groups or people already opposed to the E.U but also all over TV and mainstream media. The whole thing with Italy being left alone in dealing with the crisis made angry not only M5s or Lega Nord supporters but even supporters and members of more "traditional and moderate" parties like PD and Forza Italia.
And this is not because Italians are naturally Euroscpetic, around 54 of italians in the polls are in favour of the idea of a federal and more democratic Europe with an elected "president of Europe" and when the E.U was formed around 75% of italians were in favour of it. Not only that we also were one of the original founders of the E.U and of the other organizations that preceeded it but how the E.U has ben run so far and the state of the E.U right now left many italians very dissapointed(even among those that like the idea of a more united europe)
Many Italians feel that we are treated like second or third class members even thought we are one the mayor E.U countries in terms of economic and population size and we were one of the original founders. Now to be fair part of this responsability it's also ours and of the Berlusconi governments that made us look like clowns internationally and didn't assert themselves enough in the E.U (also because Berlusconi lacked credibility after all the gaffes and diplomatic missteps he caused) but we have also kind of been neglected a lot of the time on some issues.
Also a lot of the rethoric of the german press and of some german politicians(Schäuble) has been higly disliked by italians from nearly every single italian political formations. The rethoric of lazy Italians/Spaniards/Greek/Mediterraneans trying to leech off germany monies(when Italy has instead alway been a net contributor to the E.U and never taken any bail-out by anyone, neither by the E.U or Germany). The being categorized as P.I.G.S(It's quite clear that it's not simply random casual acronym and that is quite derogatory), comments from the Eurogruop like this one from Dijsselbloem ( http://www.dutchnews.nl/news/archives/2017/03/eurogroup-chairman-dijsselbloem-in-hot-water-over-drink-and-women-comments/) etc. etc.
Before anyone on this sub accuses me of being an an E.U hating far right posters I want to make clear that I'm actually fairly pro E.U (Or at least pro european integration not necessarily pro how the E.U is run right now) and some of my top rated comments here have actually been defending the E.U but still I think that if things continue to go this way don't be surprised if Eurosceptics take power in Italy.
I would have deemed this higly unlikely just some months ago but with how the general public felt abandoned by the E.U in the last months and weeks in regards the the migrant crisis I see it as more likely now.
This is an Italian poll from march 25 about the things that I talked about: http://www.quotidiano.net/cronaca/italia-euroscettica-1.2992667
It's all in Italian but you can use google translate to understand some of it.