What It Actually Says:
Those who insist that others should believe in God have the burden of proof.
Bertrand Russell was a noted philosopher, mathematician, and atheist. Russell rejected the Christianity he was born into and argued that those who insisted on the existence of God to others had the burden of proof. To explain why those making assertions should prove their theories, he gave an analogy about a teapot in space:
Many orthodox people speak as though it were the business of skeptics to disprove received dogmas rather than of dogmatists to prove them. This is, of course, a mistake. If I were to suggest that between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be revealed even by our most powerful telescopes. But if I were to go on to say that, since my assertion cannot be disproved, it is intolerable presumption on the part of human reason to doubt it, I should rightly be thought to be talking nonsense.
But do you see the context of the statement? It's a reaction. It's a reaction to orthodox, dogmatic people. Personally for Russell, even if he disclaimed all religions, it's a reaction to the Church of England. It's a reaction to those who want to proselytize and tell you you're going to hell unless you agree with them. It says to those people, "Hey, back off. I would never have the audacity to insist that you believed something stupid about a teapot."
I fully agree with Russell on this, but I hate the way it's used.
How It's Used Now:
Believing in God is as stupid as believing in outer space teapots!
Most of the time you see this idea referenced without the inclusion of it being a reaction to those who are "orthodox" or "dogmatic" and who find the questioning of their beliefs "intolerable." Instead, the Internet reduces the argument to something along the lines of ...
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"If you believe in God, you might as well believe in outer space teapots, derpity derp derp."
There's a reason people online get it wrong: The Internet assumes it's a fight. It assumes any person of faith wants you to believe as they do. Wants to prove you wrong. Wants to damn you for not reading from the same holy books. And if you find those people who are telling you you're going to burn in hell because there's something wrong with you for not accepting the blood of Christ or whatever faith is being preached, you'd be in your rights to cite to Russell. You could say, "Hey, don't tell me there's something wrong with me for not believing as you do when it's your job to support your incredible claims before attacking someone for not being of a similar mind."