Note: I like to read about Rust but don't work with it seriously, and don't follow Swift at all. Corrections welcome. That said, these seemed like the key passages:
"Swift has always considered read/write and write/write races on the same variable to be undefined behavior. It is the programmer's responsibility to avoid such races in their code by using appropriate thread-safe programming techniques."
"The assumptions we want to make about value types depend on having unique access to the variable holding the value; there's no way to make a similar assumption about reference types without knowing that we have a unique reference to the object, which would radically change the programming model of classes and make them unacceptable for the concurrent patterns described above."
Sounds like a system in the vein of rust but more limited, with more runtime checks and no lifetime parameters, falling back to "programmer's responsibility" when things get hard. The last paragraph makes it sound like one of the motivations is in enabling specific categories of optimizations, as opposed to eliminating races at the language level.
One of my biggest questions as a reader is how a language like C handles these cases that Swift can't handle without these guarantees. Is this a move to get faster-than-C performance? Does C do these optimizations unsafely? Is there some other characteristic of Swift that makes this harder than C? Closures get a lot of focus in the article...