I'm assuming that your instance is hosted by Adobe.
1. You need to have administrator access to the instance. At the time of instance provisioning, Adobe will create the first admin operator for you and share the credentials with the super user mentioned in contract. That super user can further create more admins on the system.
Admin user is from customer who can share the same with partner or create a new one for partner.
2. CSM stands for Customer Success Manager. Customer should be aware of this individual but in case it is not, they can reach out to Sales representative interacting with them as part of initial deal signing and get relevant details.
3. Initiate a call or email conversation with CSM to undestand how the contract can be amended to have this clause for log access. We will need a business justification for such regular access to logs.
Someone who has access to Monitoring tab of Campaign can read these logs.
Log files are rolling in nature and hence once the size reached to 10MB it rolls over.
If your instance is Adobe hosted, you can request support to provide you the logs for a specific duration but if you need it regularly, better to go through CSM and get a process in place. Please note for Adobe hosted instances, logs are purged regularly.
If you have the instance hosted on-premise, you can fetch the logs from installation folder of Campaign.