In short, I am discovering it obscure what "methinks" and "protest to much" necessarily mean here, and who "the lady" refers to. It's used (within a humourous way) when someone is earning denials of one thing so strongly that it implies that they are not telling the truth. Problematically, http://whyscientologyisbad50258.livebloggs.com/5363627/methinks-doth-protest-too-much-meaning-can-be-fun-for-anyone