In general, Rust allows you to write software that doesn't break silently. That's a quite good value proposition for large scale software, where other languages often require programers to be super careful with refactorings, while in Rust you can really refactor all the things.
The reason people are afraid to do large refactorings in say C++ is often "security-related": fear of introducing segfaults, memory errors, undefined behavior, threading errors, etc. but one can also see these fears as "productivity-related" (hours and hours of debugging), or through many other lenses (shipping bugs to users, having a segfault in the middle of a demo that costs you a client, introducing a segfault one day before the release of your game, etc.).
In general, every programmer wants to have a certain degree of security that their software "works" for some definition of "works". For some this security might be actual network security, but for others it might mean that they don't want their data-science app that has been running for a week burning thousands of dollars to crash due to a segfault while writing the results.