My computer recently started dialing out to my ISP everytime I turn it on. It also dials while in Standby mode, or randomly when I'm working offline.

The only recent download I did was to upgrade my McAfee Security Suite. I spoke with McAfee and went through procedures to deactivate the software and see if the computer still dials out, and it did. So McAfee ruled out their software as the cause.

Is there a way to tell what application is accessing the internet automatically?
And how do I stop it?

You could monitor your Task Manager / Processes tab and see what (if any) process activates CPU usage that coincides with this behavior.

You can also selectively eliminate startup programs via start Run... msconfig | Startup. Here you can experiment with shutting off programs from starting. None of these are vital to Windows: they are all third party software.

Also check your e-mail program for such options or turn it off.