In what sense does the security model of proof-of-stake require users to "log on to the internet every few months"
What Vitalik is talking about is the Slasher algorithm that he designed which punishes the block signer if he attempts to create a fork in the blockchain. However the way that slasher works is that there will be basically a timeout; if a fork is detected during that timeout, the signer that created the fork loses his money. But after the timeout, he does not. This is so that he can actually spend the money at some point.

The problem with this is that once the timeout has passed, the signer can then create a fork without being punished. This is known as a long range fork. This fork, if it survives, can grow and, at some point, could be longer than what you have synced up to. In that case, because of how PoS works, you could end up switching to using the forked chain instead of the real chain. So to avoid that, you need to go online routinely to keep your blockchain up to date so that in the event of a long range fork, you are still using the correct blockchain.