Whoever makes a claim carries the burden of proof. If I assert that a teapot orbits the sun between Earth and Mars, it is not your job to disprove it — the absence of your disproof doesn't make it likely. Demanding that others disprove an unsupported claim is a rhetorical trick that reverses the normal direction of evidence, and it's the engine behind most conspiracy reasoning.
The related error is the argument from ignorance: treating 'this hasn't been disproven' as evidence it's true, or 'we don't have an explanation' as evidence for a specific explanation. Not knowing what caused a light in the sky is not evidence for aliens; it's evidence you don't know. And the burden scales with the claim — ordinary claims need ordinary evidence, but a claim that would overturn well-established knowledge needs proportionally stronger support, since the existing knowledge is itself evidence against it.
Bertrand Russell was asked in 1952 to write about whether God exists, and he built his answer around a deliberately absurd example. Suppose, he said, I claimed that a china teapot orbits the sun somewhere between Earth and Mars, too small to be seen by any telescope. Nobody could disprove it. But, Russell argued, if he insisted that his claim therefore could not be doubted, this would rightly be considered nonsense — 'the burden of proof lies upon me' precisely because he made the assertion. His point was that unfalsifiable claims don't gain credibility from being unfalsifiable; they lose it. Carl Sagan later sharpened the principle into the line most people know: 'extraordinary claims require extraordinary evidence.' Sagan's version adds the crucial scaling idea — the strength of evidence required isn't fixed but rises with how much the claim conflicts with what's already well established.