'Significant at p<0.05' does not mean 95% likely to be true. Whether a positive result is real depends on how many of the hypotheses you test are true in the first place — the base rate. Test a field of mostly-false ideas and most of your 'significant' hits are false alarms, no matter how careful the p-value.
Over a field of hypotheses, a fraction π are true. Testing at significance α with power (1−β) yields true positives = power·π and false positives = α·(1−π); the false-discovery rate among the 'significant' results is α(1−π)/(α(1−π)+power·π), and P(true | significant) is its complement — all exact, computed live. So p<0.05 gives 95% confidence only when the base rate is high; test mostly-false ideas and a majority of your discoveries are false, exactly as the readout shows.
A single-test framing (real research runs many correlated tests, and p-hacking makes effective α worse); π is a modelling assumption, not a measured quantity. The Bayesian point — significance without a base rate is not a truth probability — is exact and is the whole lesson.