Go to Edit > Audio Hardware Setup and ensure that the box that says Release ASIO Driver in Background is not checked.
If it is checked, Audition will stop playing or (even more catastrophically) will stop recording when you "click away" from the program, and it loses focus. This enables other programs to use the ASIO driver, which normally cannot otherwise be shared, with the following exception:
Depending on the sound device you have, you can often share out ASIO channels between applications. If you want to do this, then definitely make sure that the driver isn't released, or it won't happen - obviously. Audition only "hangs onto" the ASIO channels that are actually allocated to tracks, and this applies to a lot of other applications too. Whether all ASIO sound device drivers have this capability is another matter, though - YMMV.
Please note that it is inherent in the ASIO standard that you can only have one functional ASIO hardware sound device connected to your computer - this has nothing at all to do with Audition. If you want to run more than one sound device, use ASIO4ALL and WDM or MME drivers.