One use case is for retrieving similar documents, such as when recommending related content. Another use case is retrieving document segments that are similar to a user query and passing them along with the user query to a large language model for improvement in the generated response. Vector search is also better in some ways than keyword search since it can find documents that are semantically similar even when the user may not have used the exact keyword, or even partial keywords like “Postgres” instead of “PostgreSQL.”

You can generate "text embeddings" (ie vectors) from your text with an embedding model. Once you have your text represented as vectors, then "closest vector" means "more semantically similar", as defined by the embedding model you use. This can allow you to build semantic search engines, recommendations, and classifiers on top of your text and embeddings. It's kindof like a fancier and fuzzier keyword search.

I wouldn't completely replace keyword search with vector search, but it's a great addition that essentially lets you perform calculations on text

I put together an extensive guide to understanding embeddings and vector search last year: https://simonwillison.net/2023/Oct/23/embeddings/