In philosophical discourse an existence proof is simply an assertion that a thing can exist, because it self-evidently does exist. It is not a proof in any formal, mathematical sense. It is, however, logically incontrovertible.
Is there any value in invoking an existence proof? Yes, existence proofs are typically used to make the point that something would be regarded as a laughably fantastical notion were it not for the small but crucial fact that it already exists. A good example is life itself, which arguably contravenes the second law of thermodynamics in the sense that life appears to be an instance of negative entropy - order from chaos. The emergence of organic living stuff from inorganic non-living stuff is therefore an existence proof that such a thing can happen even if we don't understand how it happens.