Heartbleed included memory safety violation. A hand crafted Malloc replacement was used to manage buffers. Uninitialized memory of new buffers was able to be read which allowed the extraction of secrets.

Now the reason why some people claim that Rust would not have stopped it, is that a similar leaky buffer system is possible to be written in Rust. This however is just applicable if you just use byte buffers for everything. At that point you become close to emulation. Also if the system malloc is shit, you just use jemalloc which is the standard for Rust. So there is no need to roll your own malloc replacement in Rust