RStudio Server runs on the same machine that you run your R code on and a web server is included. This allows RStudio Server to administer and manage the R processes it starts and stops and avoids a network hop. If you are using a database, that is a separate tier. RStudio Server itself does not require a database but does store its information on the file system.
This is also true of RStudio Workbench / RStudio Server Pro, except that a database is included in the installation for information storage.