If you are using C or C++, it is probably because you have to - either you need low-level access to the system, or need every last drop of performance, or both. Rust aims to do offer the same level of abstraction around memory, the same performance, but be safer and make you more productive.
Concretely, there are many languages out there that you might prefer to use to C++: Java, Scala, Haskell, Python, and so forth, but you can't because either the level of abstraction is too high - you don't get direct access to memory, you are forced to use garbage collection, etc. - or there are performance issues - either performance is unpredictable or its simply not fast enough. Rust does not force you to use garbage collection, and as in C++, you get raw pointers to memory to play with. Rust subscribes to the 'pay for what you use' philosophy of C++. If you don't use a feature, then you don't pay any performance overhead for its existence. Furthermore, all language features in Rust have predictable (and usually small) cost.
Whilst these constraints make Rust a (rare) viable alternative to C++, Rust also has benefits: it is memory safe - Rust's type system ensures that you don't get the kind of memory errors which are common in C++ - memory leaks, accessing un-initialised memory, dangling pointers - all are impossible in Rust. Furthermore, whenever other constraints allow, Rust strives to prevent other safety issues too - for example, all array indexing is bounds checked (of course, if you want to avoid the cost, you can (at the expense of safety) - Rust allows you to do this in unsafe blocks, along with many other unsafe things. Crucially, Rust ensures that unsafety in unsafe blocks stays in unsafe blocks and can't affect the rest of your program). Finally, Rust takes many concepts from modern programming languages and introduces them to the systems language space. Hopefully, that makes programming in Rust more productive, efficient, and enjoyable.
I would like to motivate some of the language features from part 1. Local type inference is convenient and useful without sacrificing safety or performance (it's even in modern versions of C++ now). A minor convenience is that language items are consistently denoted by keyword (`fn`, `let`, etc.), this makes scanning by eye or by tools easier, in general the syntax of Rust is simpler and more consistent than C++. The `println!` macro is safer than printf - the number of arguments is statically checked against the number of 'holes' in the string and the arguments are type checked. This means you can't make the printf mistakes of printing memory as if it had a different type or addressing memory further down the stack by mistake. These are fairly minor things, but I hope they illustrate the philosophy behind the design of Rust.