You "don't care," I'm sure, about all kinds of things when you're programming - depending on what level you're at. Unless you're drawing up CPU schematics, there is something that you have abstracted away as no longer your concern.
Rust allows you to not care about more things than C/C++ despite compiling to a comparable binary. In particular, Rust makes refactoring and extending a code base much more care-free than many other languages (including high level languages) because of the strong type and memory safety guarantees of its semantics.
EDIT (re: your edit) - No one said anything about preventing arbitrary logic errors, just that what Manishearth did will not introduce logic errors because of Rust's guarantees. This could not be the data that Manishearth wants, but the compiler guarantees that Manishearth cannot change the data or delete the data and that the data is not changed or deleted while he is referencing it. That is what is so great.