In the beginning of The Taming of the Shrew, the Lord has his page dress up as a woman:
Sirrah, you go get Bartholomew, my page,
And dress him in all suits like a lady.
That done, escort him to the drunkard's chamber,
And call him "Madam", do him obseisance.
I get why it was necessary to have a wife there; it improves the ploy. But why did they need to have Bartholomew dress up? Couldn't they have gotten a real woman to do it?
Because it's funny. A man in a dress is always funny. Or at least, it was in Elizabethan times, and was true until perhaps the last decade or so. (Now, it just looks kinda homophobic/transphobic.) The whole goal is to play a prank:
I long to hear him call the drunkard husband,
And how my men will stay themselves from laughter
When they do homage to this simple peasant.
If it's funny to trick the man, it's even funnier to trick him with a boy.
Shakespeare used the man-in-a-dress to comedic effect elsewhere, most notably Thisby. It makes a man look small and weak, and thus comic in the Aristotelian sense:
Comedy, as we have said, is a representation of inferior people, not indeed in the full sense of the word bad, but the laughable is a species of the base or ugly.
Flute tried to assert his manliness:
FLUTE: What is Thisby? a wandering knight?
QUINCE: It is the lady that Pyramus must love.
FLUTE: Nay, faith, let me not play a woman; I have a beard coming.
Another example is the Nurse in Romeo and Juliet. There's great comic effect to be mined out of a large man playing the Nurse: just as the portrayal makes the man look un-masculine, the nurse looks un-feminine. And so when she makes coarse jokes ("Now, by my maidenhead -- at twelve years old."), the effect is enhanced. Falstaff, sneaking out dressed as a woman in Merry Wives is a similar gag.
Now, this is a very careful comic balance for Shakespeare to walk, because he also played women in serious ways. He had not just great dramatic female roles like Cleopatra and Tamora, but great female comic characters for whom femininity is crucial. This is most visible in "cursed" characters like Beatrice in Much Ado or Kate in Shrew: the roles aren't funny if the audience doesn't fully accept them as attractively feminine.
The audience would, of course, have been aware that these characters were portrayed by boys. Several of Shakespeare's greatest comic female characters dress as boys: Rosalind, Viola, Julia. Again, there's a delicate balance where the audience is being asked to accept that these are boys-playing-girls-dressed as boys.
It worked only because the audience was fully imbued in the trope, and simply accepted it. I think of it as similar to the effect of intercutting in modern cinema: the world doesn't jump perspectives like that, but since you've been trained to accept that as the language of cinema, it doesn't give you even a moment's pause. It's not so much "suspension of disbelief" as genuine belief: that's the way this world works.
Of course, they only believe it in the theater. They don't accept boys-dressed-as-girls in real life. Unless, of course, they're drunk. So the effect of using Bartholomew is to make the drunken guest seem more despicable, and thus funnier.
It also helps that this keeps the prank as a "boy's club" moment. Women, especially young women, were usually portrayed as more refined in their sensibilities. Even low-born characters like Audrey in As You Like It (" I am not a slut, though I thank the gods I am foul") have some touch of propriety. Women do participate in pranks (like Mistress Quickly in Henry IV, part 1), but are usually less coarse about it. I'm certain Shakespeare could have written a woman into the scene without breaking the drunken-fraternity nature of the the prank, but it was certainly in character to make it an all-male moment.
