How does Brouwer's fixed point theorem relate to Walrasian equilibrium?
In standard proofs of the existence of Walrasian equilibrium, after one applies the appropriate fixed point theorem (typically Kakutani's, but sometimes Brouwer's), one then shows that the given fixed point is, in fact, a Walrasian equilibrium.

Depending on the exact construction of the proof, sometimes the fixed point will only give you the equilibrium price vector. However, this will be done so that the implied demands will then clear the market.

In other proofs, like this one, the fixed point will give you both the equilibrium price and allocation.