Deep copies for classes owning heap memory

Why does a C++ class that allocates memory itself need hand-written copy construction and copy assignment?

Because the defaults copy the pointer, not the data. Two objects then share one buffer: changing one alters the other, and both destructors free it (a double delete); default assignment also leaks the target's old buffer. Implement deep copies (copy-and-swap handles self-assignment and exception safety) following the rule of three/five, or better, hold the memory in a std::vector or smart pointer.