Harikatha Kaalakshepam is an art form that tells stories and sermons through Carnatic music. One of the biggest proponents of this art form in today's world is Vishakha Hari. It's only recently that I had a chance to listen to some of the discourses by Vishakha. Vishakha is a well-read lady and is a Carnatic vocalist as well. 
Let me explain what Harikatha Kaalakshepam is in a layman's language. Imagine a typical Indian movie where songs are inserted for different situations. Now, extrapolate this idea to the stories from Indian mythology where different Carnatic compositions (by the Trinity and other Vaggeyakaaras) are added for various circumstances. That is Harikatha for you. This is what I was able to comprehend from my limited exposure to this art form. 
While several resources on Carnatic music emphasize more on the melodic, rhythmic, and improvisational aspects, Harikatha emphasizes on the lyrical and contextual aspects thereby, making Carnatic compositions more meaningful and relevant to the masses. 
I need to dig more into this art form using the available resources to find more insights. For now, I will leave you with this video.

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