ASP.NET MVC is different from traditional .exe during development, in that every web request is essentially a clean run through the stack. Therefore, you don't really have to have your app "running" in visual studio, unless you are actively trying to monitor the call stack on a particular function call.
In most situations, you are making changes and then testing those changes in a browser, through IIS Express. If you start your app using CTRL+F5, it spawns a process of IIS Express using your current .DLL in the bin/debug directory, but leaves visual studio in edit mode. If you make changes to your code and then compile using F6, the .DLL in the bin directory is updated, and the next browser request made to IIS express will use the new codebase. You can continually use F5 in the browser, observe the results, make changes, recompile, etc. without breaking your workflow.
It is only necessary to run Visual Studio in debug mode if you are actively trying to debug a method call and need to set breakpoints in the server code. Making changes to the razor views, adding new controllers/actions, etc. do not generally require you to debug. And in many cases, simply using console logging or other visual cues in your HTML / Razor can be used to trace variable states and further avoid the need to rely on server debugging.
ctrl + F5 is "start without debugging" by default, which will spawn a copy of IIS express in the system tray, and launch a browser. even if you close your browser window, the IIS express instance will still be running in the tray until you close visual studio. F6 is just to recompile the current code, and would only result in a quick "build successful" status message. While IIS Express is active in the system tray, multiple browser instances can all make requests to the port which is assigned to that server, without you needing to do anything else in Visual Studio. Recompiling the code doesn't affect the current status of any browser instances currently open, but immediately affects any future actions taken from any browser window.