Not coming from a Telephony background I didn't really understand how legacy PBX systems provided voicemail. It turns out voicemail is considered a separate solution. Many PBX companies left voicemail to third party companies.
To me this seems strange but hardware was much different in the early days and mass storage wasn't as cheap as it is today. In any case there many legacy PBX systems that are still running today.
One way Asterisk can benefit from this using Asterisk as replacement voicemail server. This is good because it gets rid of some of the problems of initially trying to sell Asterisk as a complete replacement to their existing system.
Here are some reasons why Asterisk will make a good voicemail server.
1. No expensive wiring or handsets to consider (client is happy)
2. Extends life of current system (client is really happy)
3. It will add additional features than the voicemail system its replacing, vm to email for example
4. Increase their voicemail storage
5. Client will be more open to move to a full Asterisk system once they are comfortable with it being a VM server first
For some systems it may not be possible to actually integrate Asterisk. But there are systems with SMDI and sip trunking support.