There's two things going on here: 1) The "Right" refers to conservative political parties. In America, that means the Republicans, the Tea Party, etc. I'm not sure what that means in the context of the UK, though. 2) The "Weirdo Right" is just making fun of people who are on the "right" side of the political spectrum. Either that or it's talking about a subgroup of people on the "right" who have weirdo policies that don't make sense. That's about how much I can explain as an American, haha.

'Weirdo Right' is not an accepted term, but it's clear why it's being used here. The Daily Mail is a "popular" right-wing publication. The article is ridiculing two groups of people whose opinions are different from those of the Daily Mail - Labour MPs (left wing politicians) and weirdo ( i.e strange, extreme and mad) right-wingers. The Mail ( mainstream right) is disassociating itself from those right-wingers whose opinions it disapproves of by using the judgemental and pejorative term 'weirdo'.