The fact that they cannot be reused in a single page is what makes IDs so useful.
There are benefits of IDs others mentioned, but for me properly used IDs make CSS so much clearer. An ID in CSS code is a flag saying: "custom-crafted for this part of the page only, not to be reused", or "non-modular code".
Unlike a class, an ID informs me with 100% certainty that if modified, it will not affect other instances on the page, because there are none. It's hard to overestimate the value of such information, especially when working on older or someone else's code.