First some context: llm "prompts" are actually the whole conversation + initial context. They learn nothing, hence the whole conversation gets fed into them every time, but the instruction following ones are trained to answer your most recent chat response.

In a nutshell, part of your llm prompt (usually your most recent question?) gets fed as a query for the embedding/vector database. It retrieves the most "similar" entries to your question (which is what an embedding database does), and that information is pasted into the context of the llm. Its kinda like pasting the first entry from a local Google search into the beginning of your question as "background."

Some implementations insert your old conversations (that are too big to fit into the llm's context window) into the database as they are pushed out.

This is what I have seen, anyway. Maybe some other implementations do things better.

Any database can be used to extend the memory of LLMs. What a database does is store stuff and lets you search/retrieve stuff. Embeddings are differet form of data that are in many (but not all) cases superior to searching through text.

You do not need a fancy cloud hosted service to use an embeddings database like you do not need one to use a regular databse (although you could).

Check https://github.com/kagisearch/vectordb for a simple implementation of a vector search database that uses local, on-premise open source tools and lets you use an embeddings database in 3 lines of code.