It means it is more often than not, it's better to listen to people instead of constantly talking. It points out that by listening, you learn a lot more than if you were the only one that was ever talking in a meeting, social event, etc. That is the overall point of the quote. If you want to analyze it literally, he means the more that a person talks, the more likely that person is to say something stupid and get fired.

I haven't heard that before, but I would imagine it suggests that listening (keeping quiet) is the safe option, whereas being brave and speaking up (and therefore potentially saying something stupid or inapproapriate or unacceptable) could cost you your job. I suspect that the coded message behind this saying is NOT to advocate keeping quiet, but to imply that to do so is a slightly cowardly approach, and that those who adopt that approach (in anything, not just work) may be regarded as timid and unworthy of admiration.

When one actually listens, one learns. Keep learning = keep your job.