While I can't find any RAW at a quick search, affected typically means that the spell effects are active on that target, in this case it would mean they've failed their save.
Interpreting this to the specific spell, consider that if you were in a Zone of Truth and were trying to tell a lie, but were unable to (because you failed your save), this would be pretty obvious to you what's going on.
Meanwhile, if you were in a Zone of Truth, and had succeeded at saving against the effects, if you tried to tell a lie, you would be able to, so you would not necessarily know something was odd.
In either case, if you're trying to tell the truth, you wouldn't really know you're in a Zone of Truth. (It might be argued that no matter how had someone tries to tell the truth, there is always a hint of a lie to it, minor embellishments etc, that the Zone of Truth would make its effects known still, had they failed their save and tried to tell "the truth")