Outside a legal context, "burden of proof" means that someone suggesting a new theory or stating a claim must provide evidence to support it: it is not sufficient to say "you can't disprove this".
For example, if I say "The Chinese government is plotting to poision our water supply" it is my burden of proof to prove this plot is actually occurring. If I can provide evidence that proves the plot exists, then it becomes the skeptic's burden of proof to disaprove my claim with facts of his own.
See also scientific method
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