Out of interest, have you actually read anything about lie detectors?
Some. Not loads. I've read that testers allege accuracy rates in the region of 90% + but that studies have revealed accuracy rates could dip as low as 60% +.
IMO.. where a person is accused of something (not necessarily a serious crime, it could be an everyday life matter which they deem to be of some gravity) and is pleading their innocence in relation to the accusation... those are the kind of situations where a person might verbally 'offer to take a lie detector test'. Their offer is effectively a plea of cast iron certainty in their own innocence. Now in everyday life, it could also be a bluff (because it is highly unlikely that the person will actually undergo such a test).
Just an observation IMO, that it may well be an attempt to 'prove' innocence, despite the legal impotence attached to such tests.