I'm reading Illusie's survey on Crystalline cohomology, and I found him talking about those $p$-adic period rings like $B_{\text{dR}}, B_{\text{cris}}$. Can anybody explain what they are and give some heuristic on why they are what they are and what they are good for? So far I've only seen them appear in comparison isomorphisms. Are they just there to make the comparisons work?

I think I found a related question here: https://mathoverflow.net/questions/62454/fontaines-rings-of-periods but answers there simply didn't say much about it.