Regarding application of ring signatures
To understand the application, it is useful to contrast this with the use of regular signatures.
Imagine Alice sends a message $m$ to Bob together with a signature $\sigma$ of $m$ under her own public key $\mathsf{pk}_A$. A regular signature scheme now has two important properties:

It's publicly verifiable, i.e., anyone can verify that the $\sigma$ is a valid signature of $m$ under $\mathsf{pk}_A$.
It's unforgeable, i.e., only someone in possession of the corresponding signing key $\mathsf{sk}_A$ could have produced a valid $\sigma$.

The combination of the two, together with the assumption that Alice has a strong incentive to keep her signing key secret¹, leads to a property called non-repudiation. This means that the recipient can prove to a third party that the Alice really did send $m$ simply by producing $(m,\sigma)$.
Now, say $m$ is something legally or morally incriminating. Then, among other things, this opens Alice up to blackmail. Bob can threaten to release $(m,\sigma)$ which would prove to the public or some third party that Alice said $m$.
Alice understandably would like to prevent that. So this is where ring signatures come in. If we replace the signature scheme above with a ring signature scheme, then Alice would be sending $m$ together with a ring signature $\sigma$ of $m$ under the set² of her own public key and Bob's public key $\{\mathsf{pk}_A,\mathsf{pk}_B\}$.
A ring signature is still publicly verifiable, however the unforgeability now only guarantees that the signer must have known one of $\mathsf{pk}_A$ or $\mathsf{pk}_B$. This means that Bob can no longer prove to a third party that Alice sent $m$, since Bob himself could have simply computed $\sigma$ himself.
He can of course still try to blackmail Alice, but he can no longer rely on the security of the signature scheme to do so.

¹ This is where this argument could break down in practice.
² Sometimes also called ring, even though the name ring signature actually comes from the way the first such scheme computed it's signatures.