Open-source means that the source code is available for anyone to read. (It's also closely connected to copyleft licenses, which explicitly allow copying and derivative works, but they aren't quite the same.) It's generally very good for security software to be open-source, because it means anyone can see how it works and highlight security flaws, and it makes it much harder for malicious code (i.e. a password manager that sends all your passwords to the creator) to sneak in under the radar.

Sorry. If was a real writer instead of just an internet ranter, I would have either clarified the term or used different ones.

Open source software works something like Wikipedia. It's not perfect, but the fact that anyone can examine what's written and contribute means that eventually the errors get sorted out.

What I should probably have said is something more along the lines of "... one of the password manager apps that are frequently reviewed and audited for privacy, reliability, and correctness, like Bitwarden or KeePass. Both of those, and the vast majority of trustworthy password managers are available for virtually every platform."