I think the 10X web site explains how they work -- see the FAQ and Guides (10xmanagement.com).
They do act as my agent -- their model is the same as entertainment and sports agents. I pay them a percentage of my gross. They do all of the marketing, contract negotiation, legal, billing, deal with customers and get additional resources as necessary. They put projects in front of me, I get to decide what I take on. In general I'm getting higher rates through 10X than I would get on my own, and they do things that I'm not good at or that would cost me non-billable time.
I had my own freelance business when I signed on with 10X. I had decided to live overseas and got worried about getting new customers in the US, and 10X seemed like a good way to maintain and grow my freelance business.
See https://www.newyorker.com/magazine/2014/11/24/programmers-pr.... The article mentions me.
Recruiters and placement firms get paid by the employer, so that's who they work for. I have worked with placement firms who place contractors. Some of them act kind of like agents, others hire the contractor as an employee and then outsource to companies who need people and skills. All kinds of arrangements are possible, it's kind of a seller's market in IT right now, at least in the US, for people who have in-demand skills and experience.