How and why do Shakespeare's use of language differ at different times of the play?
Ooh, this is a detailed question. I'll give some overviews, and some places for you to look, and you'll have to follow up on this for yourself.
Broadly speaking, Shakespeare's language shifts from poetry to prose (and back), and from high to low diction (often with the poetry to prose shift, but not always). He also shifts the type of imagery used.
He does this to distinguish between characters and among classes of characters, and to characterize people through their speech.
The easiest examples of these contrasts come between the Oberon and Titania and the rude mechanicals. In the first speeches by Oberon and Titania, Titania speaks lovely verse. She uses many images, often sensual images, often nature images, and classical allusions.
The first time the mechanicals meet, by contrast, they speak either all in prose or in bad verse (look at Bottom's speech in Act I, scene 2).