When a Rust program is
faster than the matching C program, it is utterly nonsensical to attribute its speed to C.
It is gratifying to see C++ identified, here, as the hands-down fastest implementation language, but odd to see Rust performance still compared, in the headline, to C, as if that were the goal. The headline should say that Rust speed is approaching C++ speed. In principle, Rust speed should someday exceed C++'s, in some cases, because it leaves behind some boat anchors C++ must retain for backward compatibility. In particular, if compiler optimizers could act on what they have been told about language semantics, they should be able to do optimizations they could not do on C++ code. As it is, Rust relies on optimizers coded for C and C++.
Rust may never reliably beat C++, because Rust has itself committed to details that interfere with optimization, principally in its standard library: The C++ Standard Library offers more knobs for tuning, while the Rust libraries are simpler to use.
Some of the reasons that C++ is so reliably faster than C are subtle and, to some, surprising. As noted elsewhere in this thread, the compiler knows more about what the language is doing, and can act on that knowledge, but C and C++ compilers share their optimizer, so that makes less difference than one might guess. Mainly, the C++ code does many things you might do in C, but in exactly one way that the optimizer can recognize easily.
The biggest reason why C++ is so much faster than C is that C++ can capture optimizations in libraries and reliably deliver those optimizations to library users. The result is that people who write C++ libraries pay a great deal of attention to performance because it pays, and that attention directly benefits all users of the library, including other libraries.
Because you can capture more semantics in C++ libraries, C++ programs naturally use better algorithms than C programs can. C programs much more frequently use pointer-chasing data structures because those are easier to put into a C library, or quicker to open-code, where the corresponding C++ program will use a library that has been hand-tuned for performance without giving up anything else.
Rust gets to claim many of the same benefits, because it is also more expressive than C, and Rust libraries are often as carefully tuned. Rust is not as expressive as C++, yet, and has many, many fewer people working to produce optimal libraries, but it is doing well with what it has.