The most interesting part of this disclosure is:

    Internal stdio buffering is the most severe of the three problems
    discussed in this section, although GNU/Linux is not affected because
    the glibc mmap()s and munmap()s (and therefore cleanses) stdio buffers.
    BSD-based systems, on the other hand, are severely affected because they
    simply malloc()ate and free() stdio buffers.

Basically the exploit reveals a big chunk of memory where, very often, your private keys happen to live. The memory has been freed, but on BSD systems the memory that was freed has not been zeroed in any way, so you still see the private keys.