Many languages that ofter the kinds of memory safety guarentees that Rust does have a single allocation strategy: objects live on the heap, live for as long as they are needed, and are periodically garbage collected. This is very straightforword both conceptually and in implementation, but has very significant costs. Such languages tend to aggressively pursue ways to ameliorate allocation costs (think the Java virtual machine). Rust supports this strategy with shared boxes, memory allocated on the heap that may be referred to (shared) by multiple variables.
In comparison, languages like C++ offer a very precise control over where objects are allocated. In particular, it is common to put them directly on the stack, avoiding expensive heap allocation. In Rust this is possible as well, and the compiler will use a clever lifetime analysis to ensure that no variable can refer to stack objects after they are destroyed.
GC is just an easy solution for retarded language designers.