Can't tell you what's happening in a closed model, but in case of open ones it's just a text stream, same "take all previous tokens, compute next token" mechanism applies. Thinking vs Response is just a state change like between a system message and a user message.

Closed models probably do the same thing internally. What is shown externally is different though: you get a summary of the chain of thought, not the thoughts itself. This is done to prevent distillation.

The latest look we had at a frontier chain of thought is probably in the Huggingface incident report - I haven't actually read it yet but I saw the BlackHat talk, and it included some snippets. The thoughts look like they are approaching neuralese. The words are still understandable but the grammar is weird, simplified. In comparison, Qwen 3.8 27b thinks in valid English.