You get a buffer that's too small for whatever data gets written there. That data (presumably controllable by the attacker) overwrites whatever other data is right after the allocated (too small) buffer; in many cases this can be leveraged to gain control of the program in ways that will depend the rest of the code around that backdoor.

That code doesn't need to be buggy - simple, correct code that traverses e.g. a linked list can be abused to gain arbitrary memory writes and reads if you can overwrite the pointers in that linked list with values under your control; and these arbitrary memory writes can be abused to gain arbitrary code execution.

Exploiting a heap overflow is not as straightforward as a stack overflow, but certainly possible, there are many real world code execution vulnerabilities that arise from a buffer overflow in heap.