What are embeddings in AI and why are they important for large language models?

I'm learning about AI, large language models (LLMs), and Retrieval-Augmented Generation (RAG), and I frequently come across the term embeddings. I understand they're used to represent text as numerical vectors, but I'm not sure how they work or why they're essential.

Can someone explain what embeddings are, how they're generated, and how they're used in tasks like semantic search, vector databases, recommendation systems, and RAG applications? A simple real-world example would be helpful.