Why does Shakespeare have Viola dress as a page in Twelfth Night?What freedom does this allow her to have?

There are a couple of reasons Shakespeare has Viola dress
as a page, one is practical and the other plot
driven.

Since the parts of young females were played by
young boys, it was more comfortable for them to dress and act like young men.  This, of
course is the practical reason.

For several of his
comedies, Shakespeare used twins.  In this case, it was better to be identical male and
female twins.  (Genetically not possible but this is a
play.)

Her reason for dressing as a male is a little weak
but it works to get her in Orsino's court where she can fall in love with him and where
she can be sent to Olivia's where she falls in love with the disguised
Viola.

Mistaken identity and misunderstanding are the keys
to "new" comedy.  Shakepeare understood this and used these elements
masterfully.