It's important to know the standard library for your language, and that's just as true with C++ as with anything else. The standard library provides significant useful functionality, defines significant useful paradigms, and is generally a very powerful tool. Without the standard library of a language, there's often very little you can usefully do.
It's important to understand the standard library even if you intend to "reinvent" aspects of it, such as its allocator mechanism, its smart pointer library, or its container types. If you don't understand it, at least to some degree, you can very easily end up with inferior home-grown alternatives. It's also important to understand from the perspective of being able to comprehend other code written in the language.