If I understand it correctly - CL will use the same voting results and validators as SL, which means that the same nodes will just produce two or more blocks at the same time. So whenever a time comes for a specific node to take action (his slot comes on) - now they produce and send 1 block. But with SL and CL they will produce and send 2 blocks, each continuing it’s own chain.
Much later, when sharding comes as a solution for scalability - then the total set of nodes could be split into multiple sub-sets where each node additionally validates and supports it’s own part of the total chain (single shard).
But for starters - just the same nodes producing multiple blocks. Cardano is Proof-of-Stake system, so it doesn’t require spending lots of time and energy to create a single block, so the same nodes could easily maintain additional chains. With some upper limit on their number, of course.
This is how I understand it. Hope it helps, and hope someone corrects me if I’m wrong or missing something.