The primary scope of TPM is to assure the integrity of a platform. In this context, "integrity" means "behave as intended". ~Wiki
I'm learning more about how the TPM performs integrity covering aspects as root of trust, chain of trust, PCRs measurements, remote attestation.
On the other hand, a cloud provider needs to enforce strict security policies, requiring additional trust in the clients. To improve the mutual trust between consumer and cloud provider, so how is the TPM related & applies to security in the cloud?
You've kind of answered your own question. TPM applies to security in the cloud by providing additional trust in the client. As to how this additional trust is provided, part of that answer is in your question as well, as remote attestation is one way of doing that. Multi-level authentication is another. TPM is meant to be used to ensure platform integrity for both hardware and software. There are more, but most importantly, TPM provides secure, hardware based storage of encryption and signing keys.