I think that his point was that a healthy attitude to take on in life is to generally take other people's words as true (thus trust), but to have mechanisms to ensure that (thus verify). The key point is that unless you have specific reasons not to trust them, then you do not set up an antagonistic relationship.

For this to work, both parties must believe that 'trust, but verify' is a valid norm.

His example about contracts is exactly that. Typically, for most 'small things', you could probably just take someone's word for it. But just in case, we're going to sign on it. As I said before, this really only works when both parties accept this as a norm. Examine trying to sign a contract with someone from a culture which places far higher value on personal honor (or however you want to put it). They might get completely insulted that you would not just 'take their word for it'.

That's something of a similar situation with employers vs employees and video surveillance at work. Clearly, most employees -don't- warrant video surveillance, and simply imposing it on them could lead to serious issues (like the culture clash above).