I decided to approach this, not from the perspective of the previous question that sparked this one, but from a more creative, brainstorming approach, with sympathy for the perspective of the person whose job is Resource Manager today, facing an organizational move to Agile, and wondering "what's my job supposed to be then?"
I'm also explicitly not using Agile as a synonym for scrum. There are other frameworks! ;)
So what might a Resource Manager do in an agile organization?
First of all, I don't think they are part of an agile team. I think this is an organizational role that is outside the teams.
Perhaps they become a traditional people manager, supporting the career growth of their direct reports, including training.
But I think a better fit would be to have a Resource Manager taking a broad view across multiple agile teams. In agile, the "resource" is more fundamentally the team than the individual.
They would be the person responsible for solving problems like "we don't have all the skills we need in the team", whether by adding team members or arranging for training, and "there are 5 teams but only 3 DBAs, what do we do".
They might be responsible for figuring out which projects should go to which teams, matching strengths to project needs. They might be responsible for deciding whether, when, and how to mix up the teams.
They could also be the person to whom the SM brings impediments that are outside the team's influence or control to resolve. They could be responsible for impediments that are affecting multiple teams.
If you leave the job description of "ensure all staff are 100% occupied" on the table, it seems there are several possibilities that could benefit the organization and the teams.