Why is it possible to produce this play without much set? What are some specific and quotes from the play that support this, particularly regarding Peter Brook?
Back in the day when Shakespeare's plays were first produced, they were all performed with a minimal set. Theatrical lighting, as we would understand it, was nonexistent; even the centuries-long established convention of a proscenium arch hadn't been developed. What the Globe and other theatres of the day facilitated was a more intimate theatrical space; the audience was not separated from the action of the stage.
No doubt that the first production of Dream relied upon the acting of the actors to convey the story, and not the setting or props. The brilliance of Peter Brooks' production in 1970 appears that he returned the play to its roots by dispensing with traditional theatrical devices that had accrued upon the play in the intevening centuries.
In V.1, Theseus comments:
The poet's eye, in fine frenzy rolling,
Doth glance from heaven to earth, from earth to heaven;
And as imagination bodies forth
The forms of things unknown, the poet's pen
Turns them to shapes and gives to airy nothing
A local habitation and a name.
The essence of the play is based upon the poet's pen; all that is needed to tell the story are the actors and a place to act.
Additionally, there are several references throughout the play referring to the actors as "shadows;" A minimal set helps to convey the ethereal nature of the play.