Voicemail will ask the user to leave a message, and will email a link to the audio recording and a speech-to-text transcription of the recording to a specified email address.
Source: Voicemail - Twilio Labs
There are lots of other little TwiML apps on this site. I just wish more actions could send email. (Twilio bought Sendgrid, after all, so this seems like it should be in their realm of expertise.)