The updated library fixes many bugs, some of which were reported long ago. There are also various improvements, including (but not limited to) those related to shared/dynamic library/object handling.
It is indeed a critical component of the OS and it won't work/boot without it. For example, DOS uses this library to load all native PowerPC software, when any application or program is opened.
AmiSSL v5 can be updated easier (no errors if it is in use) when the new version of elf.library installed (something which I fixed in advance quite a while ago, so its great that everyone can now at last benefit from it).
