> Why would you have to stay with the company after you vest?
I'm assuming they are referring to someone that _did not_ exercise their options and is leaving.
[you might already know the following based on the rest of your comment, but thought i'd add it]
With options, vesting is only half of the story; you need to exercise the options which converts them to shares. ISO options (what you get as an employee) have a PTEP (Post-Termination Exercise Period) that gives you 90 days from voluntary leave (maybe varies?) to decide whether or not you'll exercise. If you don't exercise, you forfeit them.
I believe the longer PTEPs that you hear about (5-10 years) do some ISO->NSO conversion which changes the tax situation, but at least gives you flexibility & wait for some liquidity event before committing the cash.