The "Remember me" option just creates a session cookie which the service (Facebook for instance) will be happy to accept "forever". Nowadays almost all services work by generating a session cookie which the client (your browser) will transmit over and over again during the session.
So, the "remember me" option actually "only" tells the service/server to keep accepting that newly generated session cookie forever.
If you don't select it, the service will "drop your session" (aka "log you out", and technically "stop accepting you session cookie") after some inactivity time, say something between 10 minutes (online banking services) and several hours.
As other people say, "remember me" should not make a big difference if somebody is sniffing your traffic, because you should be connecting through httpS in any case... And if you use plain unencrypted http, the man in the middle would just sniff your username/password anyways, session cookies don't make a difference! So I don't think that the statement in the quote posted in the question really points at the right problem.
And if somebody hacked your PC, as others said, you have a much bigger problem than persistent session cookies!
Two other cases to mention: if you sometimes leave your computer unattended with no "screenlocker" protection, using the "remember me" option means that you will be logged in your account immediately when you visit the page... but if not and your browser remembers your passwords you are in the same situation....
Or if you use "remember me" in a computer you don't own, like a friend's computer or an internet cafe (but that is technically identical to the 'hacked PC' case anyways :).
IMHO the biggest actual difference in using the "remember me" option, is that by being permanently and constantly logged in your (Facebook for instance) account, you provide web-trackers in other pages (say when you visit some news site) with perfect and continous tracking information! Facebook and advertising companies are extremely happy with you!