I think the transcription has been available since iOS 10 was introduced last fall. I find it very useful. The transcription capability is definitely beta. It works well enough to get the basic notion of the message at a glance. It doesn't do so well if someone leaves you a rambling message, speaks softly, or the message is full of uncommon terms/names (like a home repair service tech might leave about an appointment).
Your wireless carrier needs to offer Visual Voicemail. Here are the European carriers. O2 appears to have it in the UK for example.
http://support.apple.com/en-us/HT204040
EDIT:
Here's some description of what Visual Voicemail is.
http://support.apple.com/en-us/HT201436