Many of the odd words from Shakespeare plays invoke faith somehow. Prithee is on the lighter side of that bunch.
Prithee = Pray + Thee. It's like a prayer.
‘I pray you’ and ‘I pray thee’ used to be polite ways of asking someone for something. They have the power to do something for you or give you something, like deities tend to do. This puts you at their mercy. No wonder the word 'pray' comes from the same Latin word we get 'precarious' from.
When Cassio is drunk and ashamed, he says to Othello ‘I pray you pardon me. I cannot speak.’ He isn’t praying to God that Othello pardons him (as in ‘I pray that you pardon me). He’s begging Othello to do so (as in ‘I pray you...pardon me’).
The same thing happens when Jacques bothers Rosalind (disguised as a fellow) in As You Like It:
‘I prithee, pretty youth, let me be acquainted with thee.’
‘Prithee’ could be replaced here with ‘beg you’, ‘ask you’, ‘request of you’, and the like. If you’re raising your eyebrow at the word ‘acquainted’, nice spot. It’s a joke about lady bits. Thanks, Chaucer. More on that in a future #WordNerd installment, as well as a big, beefy blasphemy post coming soon.