Actually, Shakespeare never used this specific phrase. If he had, the gramtically correct way to say it would be "I bite my thumb at thee!"

Marginalized Orientations, Gender identities, And Intersex. it’s meant to be an all inclusive umbrella term for asexuals, homosexuals, multisexuals, trans people, and intersex people. Alternative to LGBTQIAP+
Alternative to MOGII, easier to pronounce.

It pretty much mean fuck you in lamens terms, or a different way to flip someone off back in the day

This, actually means that you want to start a fight. A looong time ago people used to "bite" their thumb at someone to show that person you don't actually care about them.

It's an old English term for "Fuck you!" Is used in many of Shakespeare's plays.

When armies took prisoners in southern Europe in the middle ages, they would sometimes be offered the chance to earn their freedom by eating a fig from a mules anus. As the captors taunted their captive, offering him this degrading method of escape, they would bite their thumbs.

As has been noted before, it has been used by Shakespeare.
A qualified guess is that the phrase, (and the hand gesture that accompanies it), symbolizes oral sex,
and implies that the insulted is gay.