There are many ways to create interactive visualizations, dashboards, and applications for business users who may not have the programming skills or interest to work in R.
Before Shiny, data analysts and scientists would write algorithms / build models in R and then work with web developers to create data visualizations for websites or with BI tool experts to convert their work into proprietary GUI experiences. These choices came with some understandable lossiness, delay, and expense (for development and/or tools), as well as BI tool visualization limitations. However, once in place, these environments could provide rich, useful data exploration experiences with drag/drop and point/click interfaces for business users.
Shiny lets R developers generate interactive data visualizations for the web directly, with as much or as little GUI richness as they choose to invest. It dis-intermediates the relationship between data scientist and business user, resulting in high fidelity iterations and a timely, high quality interactive data experience without (or with much less) web development and without the limitations or cost of proprietary BI tools.
To be clear, Shiny is not a direct substitute for Tableau, Spotfire, Qlikview or other robust BI platforms. However, for the data scientist already familiar with R, the ability to create interactive web applications rapidly from their models is unrivaled with Shiny.