How does the Signature Scheme in Picnic Works
There are a few steps involved in the design:
1. design a sigma protocol with special soundness.
2. transform the above sigma protocol to a nizkpok.
3. use the above nizkpok to prove the possession of x, such that F(0) = y, where y is the public key, x is the private key, F is an OWF.

The encryption is a PRF used to sample an OWF.

When talking about H(r,m), H is a (quantum) random oracle used in step 2 to obtain a nizkpok based on FS-transformation (Uhruh transformation).

The feature of Picnic is the use of MPC-in-the-head to obtain a sigma protocol with special soundness. This is done in few steps:
1. prover emulate a number of semi-honest execution of MPC where each party has input as a secret share of the witness.
2. prover commits to all views and states of all parties in all executions.
3. verifier open a selected subset of parties from each execution and check that the parties are behaving honestly.

Intuitively, is the prover misbehave for some parties, it will be caught with some probability; on the other hand, the semi-honest security of the MPC protocol ensures that even if some parties are opened, no information on other parties inputs (and thus the witness) is revealed.

Given the above, one can see that the challenge is the set of parties to open, and the response are the decommitments to open the views of these parties.