This phrase suggests that the speaker has been told by a reliable source, implying credibility. I have it on good authority that the test results will be released next week.
Pro Tip 2/3
Not Firsthand
The information isn't acquired firsthand, but from someone believed to be trustworthy. They have it on good authority that the event has been postponed.
Pro Tip 3/3
Not Fact-Checked
Even though the source is trusted, the information should still be considered unconfirmed until verified. I have it on good authority that our team won, but let's check the official scores to be sure.