The idea is quite simple, using the same technology available for traditional karaoke to allow users to recreate spoken word instead of singing.
It could be extremely entertaining to attempt to recreate stereotypical accents or even make impressions of people with notable voices or speaking patterns.
Entertainment would be the primary aim of the app/software but I can imagine that it might also prove valuable to people looking to work on their impressions, whether it be for acting or comedy as well as people attempting to memorize lines or speeches. It could also be great practice for learning how to read from a teleprompter while still remaining in character.
Another secondary purpose for such an app/program could be to actually help people with pronunciation when learning a new language, instead of just making fun of accents.
While doing research for this idea, I did find two similar concepts, Speech Karaoke and Accent Karaoke. Speech Karaoke does not seem to be active anymore, and it seems like they never really developed a platform, instead, they offered their services hosting "speech karaoke" activities at events and Accent Karaoke seems to have been removed from the app store, so it no longer exists. Regardless, neither of these really did what I had in mind anyway. Speech Karaoke seemed to simply show you the words of famous speeches while Accent Karaoke simply had you repeat after a recording, trying to match it.
Modern karaoke machines/apps are able to track your voice and make sure it resembles the original, this would be handy when trying to recreate a certain accent or voice.
Imagine Sing Star but instead of singing songs, you could be recreating the speech in Braveheart or learning a Morgan Freeman monologue off by heart.
I doubt that it would be difficult to convert the tech used for singing karaoke to what would be necessary for speaking karaoke.