In general, Google, Amazon and Rackspace provide the platform on which you can run your Application Server Software. "Platform" is defined as that hardware and software required to allow software like Wildfly to run, generally a "machine," an operating system and various support software and utilities. The machine can be either a virtual machine, or a real one.
So Amazon and their ilk are providing essentially what you are providing during testing: a suitable environment in which to run the software. "Suitable" in this context essentially means a virtual machine that you can tailor to your specifications, including such things as memory capacity, hard drive space, number of processor cores, and so forth. You also get facilities like DNS and MX record administration.
You would integrate the software the same way you would integrate it onto your own hardware platform or development environment. Typically you would use a remote desktop connection into the virtual server, and set it up like you would any other machine. Whether you would set up Tomcat, IIS, Node, etc. would depend on whether or not it is already installed on the target machine.