There are a couple of different ways that leak detectors work. You can replace the implementation of malloc and free with ones that can track more information during allocation and are not concerned with performance. This is similar to how dmalloc works. In general, any address that is malloc‘ed but not free‘d is leaked.
The basic implementation is actually pretty simple. You just maintain a lookup table of every allocation and its line number, and remove the entry when it is freed. Then when the program is done you can list all leaked memory. The hard part is determining when and where the allocation should have been freed. This is even harder when there are multiple pointers to the same address.
In practice, you’ll probably want more than just the single line number, but rather a stack trace for the lost allocations.
Another approach is how valgrind works which implements an entire virtual machine to keep track of addresses and memory references and associated bookkeeping. The valgrind approach is much more expensive, but also much more effective as it can also tell you about other types of memory errors like out of bounds reads or writes.
Valgrind essentially instruments the underlying instructions and can track when a given memory address has no more references. It can do this by tracking assignments of addresses, and so it can tell you not just that a piece of memory was lost, but exactly when it became lost.
C++ makes things a little harder for both types of leak detectors because it adds the new and delete operators. Technically new can be a completely different source of memory than malloc. However, in practice many real C++ implementations just use malloc to implement new or have an option to use malloc instead of the alternate approach.
Also higher level languages like C++ tend to have alternative higher level ways of allocating memory like std::vector or std::list. A basic leak detector would report the potentially many allocations made by the higher level modes separately. That’s much less useful than saying the entire container was lost.