Your development and production environments will be identical, i.e. you no longer have to worry about things breaking in production
You have full control over how your boxes are set up and where they are hosted, i.e. you are not beholden to any programming language/stack or web host (cut the middle-man)
You want to be able to spool up servers and build/deploy apps like they're going out of style (idea to execution in shortest path possible)
For Clients/Entrepreneurs:
You want your projects built faster (maximize ROI)
You want your apps to scale easily as you grow
You don't have the resources to pay for typical agency rates and hours
You want to test out your awesome new business idea but minimize your risk in terms of development effort