I would like to know what "Methinks the lady does protest too much" means in the following sentences:
‘Yep,’ says Will. ‘Lost all the feeling in the pads of these fingers.’ He holds up one hand towards me. ‘The fingerprints have gone from a couple of them.’ I squint. They don’t actually look all that different to me. And yet I find myself saying, ‘Oh yes, I think I can. Wow.’ I sound like a fangirl.
Charlie turns to me. ‘I didn’t realise you’d seen the show,’ he says. ‘When did you watch it? We’ve never watched it together.’ Oops. I think of those afternoons, setting the kids up with CBeebies, and watching Will’s show on my iPad in the kitchen as I heated up their dinner. He looks to Will. ‘No offence, mate – I do keep meaning to catch it.’ This isn’t true. You can tell from the way he says it that it isn’t true. He hasn’t made any attempt to sound genuine.
‘No offence taken,’ Will says mildly.
‘Oh,’ I say. ‘I’ve never watched the whole thing. I . . . caught the highlights, you know.’
‘Methinks the lady does protest too much,’ Peter says. He takes hold of Will’s shoulder, grinning. ‘Will, you’ve got a fan!’
- Lucy Foley, The Guest List, Chapter 12
This is a thriller novel published in 2020 in the United Kingdom. One hundred and fifty guests would be gathering at some remote and deserted fictional islet called Inis an Amplóra off the coast of the island of Ireland to celebrate the wedding between Jules (a self-made woman running an online magazine called The Download) and Will (a celebrity appearing in a TV show program called Survive the Night). The day before the actual wedding day, Hannah, the wife of Charlie (Jules' friend), arrived at the island and is now at the dinner party for the rehearsal dinner with only some selected guests. And during the party, Hannah accidentally reveals that she saw Survive the Night, and nearly fangirled Will. At this, Charlie, who has always been disapproving Will, asks his wife when she watched it. So Hannah says she didn't watch the whole thing, but just caught the highlights. Then Peter says, "Methinks the lady does protest too much."
In this part, I wonder what "Methinks the lady does protest too much" means. Is it right to understand that "the lady" here refers to Hannah, and that Peter is basically saying "Stop negating, I know you're already in love with Will!"?
In short, I am finding it difficult to understand what "methinks" and "protest to much" mean here, and who "the lady" refers to.