The chance that a positive result is a false positive depends on the false positive rate of your test and on total population statistics.

E.g. imagine your test has a 5% false positive rate for a disease only 1 in 1 million people has. If you test 1 million people you expect 50,000 false positive and 1 true positive. So the chance that one of those positive results is a false positive is 50,000/50,001, not 5/100.

Using a p-value threshold of 0.05 similar to saying: I'm going to use a test that will call a false result positive 5% of the time.

The author said: chance that a positive result is a false positive == the false positive rate.