I think this is a rather feeble joke in which a German presenter is speaking English. "Fog off" is a variant of "fuck off", meaning get lost. This could be a mispronunciation of "fuck", or a euphemism to avoid using a swearword, or a pun on the UK's undeserved reputation for being foggy.

So I can't tell you exactly what was going through his mind; I'm just giving a general idea.

And now here's the context:

Giving Britons this year’s “golden dumbass” prize, Oliver Welke, the German equivalent of Dara Ó Briain, describes how Theresa May “can’t get out of the EU and can’t even get out of her bloody car”, over pictures of Angela Merkel waiting embarrassedly outside her Berlin residency as the door of our prime minister’s limo fails to open. “Just go!” the host yells. “Hard Brexit, soft Brexit, liquid Brexit, just fog off!”
http://www.theguardian.com/commentisfree/2018/dec/25/brexit-...

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Note added at 10 hrs (2021-02-21 08:38:18 GMT)
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Apropos of nothing, Britain's island mentality is perhaps best summed up by a famous newspaper headline that probably never existed: Fog in Channel, Europe Cut Off.

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Note added at 10 hrs (2021-02-21 08:42:43 GMT)
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And Dara Ó Briain is an Irish comedian who hosts a weekly news quiz on BBC television.