While it's a fascinating visualisation, the whole thing is of course a profound and not always helpful simplification. In some countries (including Belgium), dominant parties tagged by the Guardian as right-wing are in fact pretty centrist, believing in social cohesion and strong welfare systems, just without the historical or ideological baggage that parties with the word 'socialist' or 'social democrat' in the name may have; the stark blue/red distinction is not really appropriate. In some cases (Austria in particular) there may have been a strong left-wing party in a junior coalition role at various times, but the Guardian's colour scheme would paint the whole country blue. In some cases (not to give any recent examples), one can seriously question whether a supposedly left-wing party actually behaves that way when in government.
Be that as it may, I don't think Brin's "solid shift to the right" is really correct. At the very start of the timeline, 1972, only one of the original six members of the EEC is marked in red (Germany). Throughout 1980, of the then nine member states, only two (Germany and Denmark) are marked in red. From mid-1987 to early 1988, it's only two out of twelve (Spain and Greece). The map for today includes also technical governments in Belgium, Italy and Greece, so the tally for the right is really 20 out of 27, which is proportionally less than 1987's 10 out of 12, or 1980's 7 out of 9, or 1972's 5 out of 6.
Probably the reason it looks so striking is that none of the red countries of today is big; and also for the casual viewer, today's map contrasts more recently with that of 1999, when 10 out of 15 member states had leftish governments (the exceptions, apart from Spain, being small and/or marginal - Ireland, Finland, Belgium and Luxembourg). So while Brin does have a point in suggesting that the left are at a historically weak point in EU politics right now, and that's certainly true if compared with twelve years ago, it's not really the first time that this is the case.