If you work a bit with standard Dirichlet series you see that they encode the basic operations of multiplicative number theory in a way analogous to how power series encode combinatorial constructions when thought of as generating functions. For instance, multiplying a Dirichlet series by the reciprocal of the Riemann zeta function performs Mobius inversion on its sequence of coefficients. L-functions encode multiplicative phenomena on a fundamental formal level.

Note there seem to be two versions of the former paper online: one turns up publicly available on Google, but seems to be less complete than the other version.

I'd recommend watching Youtube lectures on the basics. The literature is so daunting that if you just started reading it might be a long time before you were even conversant. I've been watching the Munster lectures on Higher Algebra on Youtube. I think they're good for preparation if you have something in mind you want to read afterwards, and it's definitely quicker than reading Higher Algebra yourself. Generally I think there are a lot of good talks on derived algebraic geometry on Youtube.

Do you know of an example in the literature which uses this construction?

Thank you, I will read your paper. The $p$-adic analytic description may indeed be helpful on its own.

I didn’t check details, but it seems like it might be possible to show directly from the generators-and-relations definition of Kahler differentials that the colimit of unramified ring morphisms is unramified.